Associated Legendre Functions and Spherical Harmonics of Fractional Degree and Order

Trigonometric formulas are derived for certain families of associated Legendre functions of fractional degree and order, for use in approximation theory. These functions are algebraic, and when viewed as Gauss hypergeometric functions, belong to types classified by Schwarz, with dihedral, tetrahedral, or octahedral monodromy. The dihedral Legendre functions are expressed in terms of Jacobi polynomials. For the last two monodromy types, an underlying `octahedral' polynomial, indexed by the degree and order and having a non-classical kind of orthogonality, is identified, and recurrences for it are worked out. It is a (generalized) Heun polynomial, not a hypergeometric one. For each of these families of algebraic associated Legendre functions, a representation of the rank-2 Lie algebra so(5,C) is generated by the ladder operators that shift the degree and order of the corresponding solid harmonics. All such representations of so(5,C) are shown to have a common value for each of its two Casimir invariants. The Dirac singleton representations of so(3,2) are included.

It is less well known that Ferrers functions P µ ν (z) of a fixed order µ, and degrees that may be non-integral but are spaced by integers, can also be used in series expansions. The fundamental relation, due to Love and Hunter [21], is one of biorthogonality: which holds if (i) Re µ ∈ (−1, 1), and (ii) the degrees ν, ν ′ differ by a nonzero even integer and are not half-odd-integers. For suitable ν 0 , µ ∈ C, this makes possible bilateral expansions of the form (1.2) and in particular, the calculation of the coefficients cn as inner products in L 2 [−1 , 1].
The simplest example is it equals unity. It satisfies differential recurrences on n and m, and three-term non-differential recurrences, as well. The function r m 0 (u) has a hypergeometric representation in the Gauss function 2 F 1 : it equals 2 F 1 −2m, − 1 4 − 3m; 3 4 − m u . But r 0 n (u), which according to (1.3), appears in series of the form (1.2) when µ = 1 4 , is less classical. It satisfies a second-order differential equation on the Riemann u-sphere with four singular points, not three; so (if n ≥ 0) it is a Heun polynomial, not a hypergeometric one. The functions {r 0 n (u)} n∈Z are mutually orthogonal on the u-interval [0, 1], in a sense that follows from (1.1), but the orthogonality is of an unusual Sturm-Liouville kind.
It is clear from (1.3) that for any n, m ∈ Z, the function P 1 4 +m − 1 6 +n (z = cos θ) depends algebraically on z, and can be evaluated using radicals. Each of the function families considered in this paper is similarly algebraic, and because any Legendre function can be written in terms of 2 F 1 , the results below are really trigonometric parametrizations of families of algebraic 2 F 1 's. To see a link to prior work, recall from Frobenius theory that each Legendre function of degree ν and order µ satisfies a differential equation on the Riemann sphere with three singular points, the characteristic exponent differences at which are µ, µ, 2ν + 1. It is a classical result of Schwarz (see [35], and for more recent expositions, [11, § 2.7.2], [33,chap. VII] and [25]) that any such equation will have only algebraic solutions only if the (unordered, unsigned) triple of exponent differences falls into one of several classes. The triples from (ν, µ) = (− 3 ) + (m, m, 2n), and they lie in Schwarz's octahedral class V.
The families treated below include octahedral ones, with (ν+ 1 2 , µ) ∈ (± 1 3 , ± 1 4 )+Z 2 , and tetrahedral ones, with (ν + 1 2 , µ) ∈ (± 1 4 , ± 1 3 )+Z 2 or (± 1 3 , ± 1 3 )+Z 2 ; the Schwarz classes for the latter being II and III. The resulting Legendre functions are octahedral or tetrahedral in the sense that their defining differential equation, on the Riemann z-sphere, has as its projective monodromy group a finite subgroup of the Möbius group P SL(2, R), which is octahedral or tetrahedral. This will not be developed at length, but there is a strong geometric reason why {r m n (u)} n,m∈Z deserve to be called octahedral functions, or (when n, m ≥ 0) polynomials. For general n, m, the lifted functionr m n =r m n (s) := r m n (u = s 4 ) turns out to satisfy an equation on the Riemann s-sphere with 14 singular points. These include s = 0, ±1, ±i, ∞, which are the six vertices of an octahedron inscribed in the sphere; and also, the centers of its eight faces.
Up to normalization, the doubly indexed functions r m n (u) are identical to specializations of triply-indexed ones introduced by Ochiai and Yoshida in their groundbreaking work on algebraic 2 F 1 's [29]. For Schwarz classes such as the octahedral and tetrahedral, they considered the effects of displacing the triple of exponent differences, not by (m, m, 2n) as in the Legendre case, but by general elements of Z 3 . It is a key result of the present paper that in the Legendre case, when the triple has only two degrees of freedom, it is far easier to derive and solve recurrences on exponent displacements.
Schwarz's classification of algebraic 2 F 1 's also includes a dihedral class (class I) and a related 'cyclic' class (unnumbered but called class O here). Legendre functions lie in class I when the order µ is a half-odd-integer, and in class O when the degree ν is an integer. We obtain explicit formulas for the Legendre (and Ferrers) functions in the respective families, of the first and second kinds. The simplest dihedral example is where m = 0, 1, 2, . . . , and α ∈ C is arbitrary. Here, P (α,−α) m is the Jacobi polynomial of degree m, and {·} α,+ signifies the even part under α → −α.
When m = 0, this becomes a trigonometric version of a well-known algebraic formula [11, 3.6(12)]; and when α = 1 2 , it expresses P 1 2 +m 0 in terms of the mth Chebyshev polynomial of the third kind. But the general Jacobi representation (1.4) is new. There is a significant literature on 'dihedral polynomials' appearing in dihedrally symmetric 2 F 1 's [29,38], and Vidūnas has shown they can be expressed as terminating Appell series [38]. Focusing on the Legendre case, when two of the three exponent differences are equal, leads to such simpler formulas as (1.4), for both the dihedral and cyclic families.
Constructing bilateral Ferrers series of the form (1.2) is facilitated by the explicit formulas derived below for the Legendre and Ferrers functions in the several families. But the functions {P µ0+m ν0+n (z = cos θ)}, (n, m) ∈ Z 2 , and the corresponding spherical harmonics {Y µ0+m ν0+n (θ, φ)}, do not fit into conventional SO(3)-based harmonic analysis unless (ν 0 , µ 0 ) = (0, 0), when the latter are the usual surface harmonics on S 2 = SO(3)/SO (2). In the octahedral and tetrahedral families (and also the dihedral and cyclic, if ν 0 resp. µ 0 is rational), it is nonetheless the case that each spherical harmonic can be viewed as a finite-valued function on S 2 . This is due both to ν 0 , µ 0 being rational, and to the algebraicity of P µ0+m ν0+n (z) in its argument z, as seen in (1.3).
To begin to relate the present results to harmonic analysis, we interpret in Lietheoretic terms the recurrences satisfied by any family {P µ0+m ν0+n (z)} or {Y µ0+m ν0+n (θ, φ)}, (n, m) ∈ Z 2 , which are based on first-order differential operators that perform ladder operations. It is well known that for any (ν 0 , µ 0 ), there is an infinitedimensional representation of the Lie algebra so(3, R) on the span of {Y µ0+m ν0 (z)} m∈Z . (In the case when (ν 0 , µ 0 ) ∈ Z ≥ × Z, this includes as an irreducible constituent the familiar (2ν 0 +1)-dimensional representation carried by the span of Y −ν0 ν0 , . . . , Y ν0 ν0 .) There is also a representation of so(2, 1) on the span of {Y µ0 ν0+n (z)} n∈Z . The real algebras so(3, R), so(2, 1) are real forms of the complex Lie algebra so(3, C).
As we explain, these 'order' and 'degree' algebras generate over C a 10-dimensional, rank-2 complex Lie algebra isomorphic to so(5, C), which for any (ν 0 , µ 0 ), acts differentially on the family {r ν0+n Y µ0+m ν0+n (θ, φ)}, (n, m) ∈ Z 2 , of generalized solid harmonics on R 3 . The root system of so(5, C), of type B 2 , comprises the eight displacement vectors ∆(ν, µ) = (0, ±1), (±1, 0), (±1, ±1), which yield four ladders on (ν, µ); and for each ladder, there are differential operators for raising and lowering, a differential recurrence satisfied by P µ ν (z = cos θ), and a three-term non-differential recurrence. The ones coming from the roots (±1, ±1), such as the 'diagonal' recurrences may be given here for the first time. (In this identity, P may be replaced by Q.) Connections between associated Legendre/Ferrers functions, or spherical harmonics, and the complex Lie algebra so(5, C) [or its real forms so (3,2), so(4, 1) and so(5, R)] are known to exist. (See [27] and [28,Chaps. 3,4], and [4,9] in the physics literature; also [19] for hyperspherical extensions.) But, most work has focused on functions of integral degree and order. The octahedral, tetrahedral, dihedral, and cyclic families yield explicit infinite-dimensional representations of so(5, C) and its real forms, which are carried by finite-valued solid harmonics on R 3 . When (ν 0 , µ 0 ) = ( 1 2 , 1 2 ) or (0, 0), the representation of so(3, 2) turns out to include a known skew-Hermitian one, of the Dirac singleton type (the 'Di' or the 'Rac' one, respectively). But in general, these Lie algebra representations are new, non-skew-Hermitian ones, which do not integrate to unitary representations of the corresponding Lie group. It is shown below that any of these representations of so(5, C) [or any of its real forms], carried by a harmonic family {r ν0+n Y µ0+m ν0+n (θ, φ)}, (n, m) ∈ Z 2 , is of a distinguished kind, in the sense that it assigns special values to the two Casimir invariants of the algebra, these values being independent of (ν 0 , µ 0 ); cf. [4,19].
This paper is structured as follows. In § 2, facts on Legendre/Ferrers functions that will be needed are reviewed. In § 3, the key results on the octahedral functions r m n are stated, and explicit formulas for octahedral Legendre/Ferrers functions are derived. These are extended to the tetrahedral families in § § 4 and 5. In § 6, the results in § 3 are proved. In § 7, Love-Hunter biorthogonality is related to Sturm-Liouville biorthogonality. In § 8, formulas for Legendre/Ferrers functions in the cyclic and dihedral families are derived, and Love-Hunter expansions in dihedral Ferrers functions are briefly explored. In § 9, recurrences on the degree and order, valid for any (ν 0 , µ 0 ), are derived, and are given a Lie-theoretic interpretation: so(5, C) and its real forms are introduced, and their representations carried by solid harmonics are examined.

Preliminaries
The (associated) Legendre equation is the second-order differential equation on the complex z-plane. For there to be single-valued solutions, the plane is cut along the real axis either from −∞ to 1 (the Legendre choice), or from −∞ to −1 and from 1 to +∞ (the Ferrers choice). The respective solution spaces have P µ ν (z), Q µ ν (z) and P µ ν (z), Q µ ν (z) as bases, except in degenerate cases indicated below.
Further light on endpoint behavior is shed by Frobenius theory. Equation (2.1) has regular singular points at z = −1, 1 and ∞, with respective characteristic exponents expressed in terms of the degree ν and order µ as +µ/2, −µ/2; +µ/2, −µ/2; and −ν, ν + 1. The exponent differences are µ, µ, 2ν + 1. The functions P µ ν , P µ ν are Frobenius solutions associated to the exponent −µ/2 at z = 1, and the second Legendre function Q µ ν is associated to the exponent ν + 1 at z = ∞. (The second Ferrers function Q µ ν is a combination of two Frobenius solutions.) These functions are defined to be analytic (or rather meromorphic) in ν, µ [30], the Legendre functions having the normalizations (For P µ ν , replace z − 1 in the prefactor on the right of (2.8a) by 1 − z; the alternative expressions (2.8b),(2.8d) come from (2.8a),(2.8c) by quadratic hypergeometric transformations.) Here, 2 F 1 (a, b; c; x) is the Gauss function with parameters a, b; c, defined (on the disk |x| < 1, at least) by the Maclaurin series In this and below, the notation (d) k is used for the rising factorial, i.e., (The unusual second half of this definition, which extends the meaning of (d) k to negative k so that (d) k = [(d + k) −k ] −1 for all k ∈ Z, will be needed below.) If in any 2 F 1 (a, b; c; x) in (2.8), the denominator parameter c is a non-positive integer and there is an apparent division by zero, the taking of a limit is to be understood.
The Gauss equation satisfied by 2 F 1 (a, b; c; x) has the three singular points Taking into account either of (2.8a),(2.8c), one sees that this triple is consistent with the exponent differences µ, µ, 2ν +1 at the singular points z = 1, −1, ∞ of the Legendre equation (2.1). Schwarz's results on algebraicity were originally phrased in terms of the Gauss equation, its solutions such as 2 F 1 , and the (unordered, unsigned) triple 1 − c, c − a − b, b − a; but they extend to the Legendre equation, its solutions, and the triple µ, µ, 2ν + 1.

Octahedral Formulas (Schwarz Class V)
This section and Sections 4 and 5 derive parametric formulas for Legendre and Ferrers functions that are either octahedral or tetrahedral (two types). The formulas involve the octahedral polynomials, or functions, {r m n (u)} n,m∈Z . Section 3.1 defines these rational functions and states several results, the proofs of which are deferred to § 6.

Indexed Functions and Polynomials
Definition 3.1 For n, m ∈ Z, the rational functions r m n = r m n (u) and their 'conjugates' r m n = r m n (u) are defined implicitly by which hold on a neighborhood of u = 0. Here, f + 108 pv = 0. Equivalently, [For later use, note that (3.1) is familiar from trigonometry as a 'triple-angle' formula: R = tanh 2 ξ if T = tanh 2 (ξ/3); R = coth 2 ξ if T = coth 2 (ξ/3); and It is clear from the definition that r m n , r m n are analytic at u = 0, at which they equal unity; though it is not obvious that they are rational in u. But it is easily checked that the Gauss equations satisfied by the two 2 F 1 (x) functions have respective exponent differences (at the singular points x = 0, 1, ∞) equal to ( 1 4 , 1 3 , 1 2 ) + (m, n, 0) and (− 1 2 ) + (−m, n, 0). These triples lie in Schwarz's octahedral class IV, so each 2 F 1 (x) must be an algebraic function of x. The definition implicitly asserts that if these algebraic 2 F 1 's are parametrized by the degree-6 rational function x = R(u), the resulting dependences on u will be captured by certain rational r m n = r m n (u), r m n = r m n (u). In the terminology of [29], these are octahedral functions of u. (ii) For unrestricted (n, m) ∈ Z 2 , r m n (u) is a rational function that equals unity at u = 0 and is asymptotic to d m n u 3n+2m as u → ∞. The functional form of the octahedral functions that are not polynomials, which are indexed by (n, m) ∈ Z 2 \ Z 2 ≥ , is not complicated.
where on each line, Π k (u) signifies a polynomial of degree k in u, with its coefficient of u 0 equalling unity, and its coefficient of u k coming from the preceding theorem.
On their indices n, m, the r m n satisfy both differential recurrences and threeterm non-differential recurrences. The former are given in § 6 (see Theorem 6.1), and the latter are as follows.
where pv, pe, p f are the polynomials in u, satisfying p 2 e − p 3 f + 108 pv = 0, that were introduced in Definition 3.1. Moreover, they satisfy which are second-order 'diagonal' recurrences.
From the first two recurrences in this theorem, one can compute r m n for any n, m ∈ Z, if one begins with r 0 0 , r 1 0 , r 0 1 , which are low-degree polynomials in u computable 'by hand.' In fact, r 0 0 (u) = 1, (3.3) By specializing to u = 1 (at which pv, pe, p f equal 0, −64, 36), one can prove by induction that r m n (1) = (−64) m+n if m ≥ 0. Examples of octahedral functions that are not polynomials because they have at least one negative index, illustrating Theorem 3.2, are . These also follow from the recurrences of Theorem 3.3.
The recurrences are non-classical, not least because they are bilateral: they extend to n, m < 0. It is shown in § 6 that for n, m ≥ 0, the degree-(3n + 2m) polynomials r m n in u are (generalized) Heun polynomials, rather than hypergeometric ones; they are not orthogonal polynomials in the conventional sense. A useful third-order (i.e., four-term) recurrence on k for the coefficients {a k } 3n+2m k=0 of r m n is given in Theorem 6.4.
An important degenerate case is worth noting: the case n = 0. For any m ≥ 0, there are hypergeometric representations in 2 F 1 for the degree-2m octahedral polynomials r m 0 and r m 0 , namely These follow by a sextic hypergeometric transformation of the 2 F 1 's in Definition 1, as well as by the methods of § 6. The first can also be deduced from the n = 0 case of the recurrence on m in Theorem 3.3. These representations extend to m ∈ Z.

Explicit Formulas
The following two theorems (Theorems 3.4 and 3.5) give trigonometrically parametrized formulas for the Legendre/Ferrers functions P µ ν , P µ ν when (ν, µ) equals (− Let hyperbolic-trigonometric functions A ± , positive on (0, ∞), be defined by . This choice is motivated by Definition 3.1: if R = R(u) and T = T (u) = −12u/(1 + u) 2 are alternatively parametrized as tanh 2 ξ and tanh 2 (ξ/3), respectively, it is not difficult to verify that the three polynomials in u that appear in Definition 3.1 will have the ξ-parametrizations (3.6c) Moreover, and more fundamentally, u = −A − /A + . Also in this section, letr m n signify r m n /d m n ; so that when n, m ≥ 0,r m n is a scaled version of the octahedral polynomial r m n , with its leading rather than its trailing coefficient equal to unity. Equivalently,r m n (u) = u 3n+2m r m n (1/u).
[Note that as ξ increases from 0 to ∞, the argument u = −A − /A + of the first r m n , which satisfies T (u) = tanh 2 (ξ/3) and R(u) = tanh 2 ξ, decreases from 0 to −(2− √ 3) 2 ≈ −0.07, which is a root of p f (u) = 1 + 14u + u 2 .] Proof These formulas follow from the hypergeometric representation (2.8b) of P µ ν = P µ ν (z), together with the implicit definitions of r m n , r m n (see Definition 3.1). If z = cosh ξ, the argument 1 − 1/z 2 of the right-hand 2 F 1 in (2.8b) will equal tanh 2 ξ. This is why it is natural to parametrize Definition 3.1 by letting R = R(u) equal tanh 2 ξ, with the just-described consequences. In deriving the formulas, one needs the representation (3.6b); and for the second formula, the definition (3.2) of d m n .
with the results in Theorem 3.4. The symmetrical forms of the right-hand prefactors are obtained with the aid of the gamma-function identities of Vidūnas [37].
Proof Combine the P →Q reduction (2.5) with the results in Theorem 4.1.
[Note that as ξ decreases from ∞ to −∞, the argument u = −C − /C + of the first r m n , which satisfies T (u) = coth 2 (ξ/3) and R(u) = coth 2 ξ, decreases from 0 Proof By analytic continuation of the results in Theorem 4.2; or in effect, by replacing ξ by ξ + iπ/2.
where expressions for the right-hand Legendre functions are provided by Theorem 4.2, hold for n ∈ Z and ξ ∈ (0, ∞).

Theorem 5.2 The formulas
where expressions for the right-hand Ferrers functions are provided by Theorem 4.3, hold for n ∈ Z and ξ ∈ (−∞, ∞).

Theorem 5.3
The formulas where expressions for the right-hand Ferrers functions are provided by Theorem 4.3, hold for n ∈ Z and ξ ∈ (−∞, ∞).
6 Proofs of Results in Section 3.1 The octahedral and tetrahedral formulas in § § 3.2, 4, and 5 followed from the theorems in § 3.1 on the octahedral functions r m n = r m n (u), which were stated without proof. The present section provides proofs, in some cases sketched, and obtains a few additional results. These are Theorem 6.1 (on the differential equation and differential recurrences satisfied by r m n ), and Theorems 6.2, 6.3, and 6.4 (on the interpretation of r m n when n, m ≥ 0 as a hypergeometric, Heun, or generalized Heun polynomial). This section also reveals the origin of the degree-6 rational function Consider a Riemann sphere P 1 s , parametrized by s and identified by stereographic projection with the complex s-plane. (As usual, s = 0 is at the bottom and s = ∞ is at the top; points with |s| = 1 are taken to lie on the equator.) Let a regular octahedron (a Platonic solid) be inscribed in the sphere, with its six vertices v 1 , . . . , v 6 at s = 0, ±1, ±i, ∞, i.e., at the five roots of qv(s) := s(1 − s 4 ) and at s = ∞. By some trigonometry [33, chap. VII], one can show that the twelve edge-midpoints e 1 , . . . , e 12 of the octahedron, radially projected onto the sphere, are located at s = (±1 ± i)/ 3)/2, which are the roots of q f (s) := 1 + 14s 4 + s 8 . The polynomials qv, qe, q f are (relative) invariants of the symmetry group of the octahedron, which is an order-24 subgroup of the group of rotations of the Riemann s-sphere.
The well-known octahedral equation states that q 2 e −q 3 f +108 q 4 v = 0. The validity of this identity (a syzygy, in the language of invariant theory) suggests considering the degree-24 rational functionR =R(s) equal to On the s-sphere,R(s) equals 0, 1, ∞ at (respectively) the vertices, the face-centers, and the edge-midpoints. It is an absolute invariant of the symmetry group of the octahedron. (Its derivative dR(s)/ds can be written as −432 q 3 v q 2 f /q 3 e but is only a relative invariant.) The covering P 1 s → P 1 x given by x =R(s) is ramified above x = 0, 1, ∞, and its ramification structure can be written as (6)4 = (8)3 = (12)2: each of the six points above s = 0 (i.e., the vertices) appears with multiplicity 4, etc.
Following and extending Schwarz [33,35], consider the effect of lifting the Gauss hypergeometric equation satisfied by 2 x , the 2 F 1 (x) appearing in Definition 3.1, from the x-sphere to the s-sphere, along x = R(s). It should be recalled that the Gauss equation satisfied by is the Frobenius solution associated to the zero exponent at x = 0.) The effects of the ramified lifting by x =R(s) are conveniently expressed in the classical notation of Riemann P-symbols, which display the exponents at each singular point [33,39]. For the 2 F 1 of Definition 3.1, one can write because any pair of characteristic exponents at a point x = x 0 beneath a ramification point s = s 0 of order k is multiplied by k when lifted. This function of s satisfies a differential equation on the s-sphere with the indicated singular points and exponents. If Lg = 0 is any Fuchsian differential equation on the s-sphere, the modified equation L ′ g ′ = 0 obtained by the change of dependent variable g ′ = (1 − s/s 0 ) α g has its exponents at s = s 0 shifted upward by α, and those at s = ∞ shifted downward by the same. As an application of this, one deduces from (6.2b) that because e 1 , . . . , e 12 are the roots of qe, and v 6 = ∞. The left-hand functionf =f (s), which by examination isr m n (s) := r m n (u = s 4 ), will be the solution of a 'lifted and shifted' differential equation on the s-sphere, with the indicated exponents. After the shifting, the edge-midpoints e 1 , . . . , e 12 cease being singular points, because the new exponents at each are 0, 1, which are those of an ordinary point.
It is straightforward if tedious to compute the differential equation satisfied byf =r m n (s) := r m n (u = s 4 ) explicitly, by applying to the appropriate Gauss equation of the form (6.1) the changes of variable that perform (i) the lifting along s → x =R(s), and (ii) the multiplication by [qe(s)] 1 12 +m+n . One finds that f satisfiesL m nf = 0, wherẽ .
That the singular points of this operator are the roots v 1 , . . . , v 5 of qv (plus v 6 = ∞), and the roots f 1 , . . . , f 8 of q f , is clear; as is the fact that their exponents are as shown in the P-symbol (6.3c). The degenerate case n = m = 0 is especially interesting. As one expects from the P-symbol, the operatorL 0 0 is simply the Laplacian D 2 s , the kernel of which is spanned by 1, s. Forf =f(s) = r m n (s 4 ), it is easy to rule out any admixture of the latter solution by examining Definition 3.1; and because r 0 0 (u) equals unity at u = 0, the base octahedral function r 0 0 must be identically equal to unity.
Because r 0 0 ≡ 1, it follows from Definition 3.1 that the hypergeometric function appearing in the definition of r m n when n = m = 0, which is x), must be algebraic in its argument x. This is essentially the 1873 result of Schwarz [35], the proof of which was later restated in a P-symbol form by Poole [33]. However, it is not trivial to extend this result on r 0 0 to a constructive proof that r m n = r m n (u) is a rational function of u for each (n, m) ∈ Z 2 . This is because the differential equationL m nf = 0, as one sees from (6.4), is far more complicated than . A constructive proof is best based on contiguity relations between adjacent (n, m), i.e., recurrences in the spirit of Gauss, derived as follows.
First, simplify the lifting along the covering s → x, i.e., along the degree-24 map x =R(s). Each octahedral functionr m n (s) turns out to 'factor through' u = s 4 , so it suffices to lift the Gauss hypergeometric equation from the x-sphere P 1 x to the u-sphere P 1 u , along the degree-6 map x = R(u) of Definition 3.1, i.e., Replacing the lifted variable s by u = s 4 quotients out an order-4 cyclic group of rotations of the s-sphere (and hence of the octahedron), about the axis passing through its north and south poles. The syzygy becomes p 2 e − p 3 f + 108 pv = 0. The roots u = 0, 1 of pv, and u = ∞, correspond to the south-pole vertex, the four equatorial ones, and the north-pole one. The three roots u = (3 + 2 2) 2 of pe correspond to the four edge-midpoints in the northern hemisphere, the four on the equator, and the four to the south. The two roots u = −(2 ± √ 3) 2 of p 2 f correspond to the four facecenters in the north, and the four in the south. The covering P 1 u → P 1 x is still ramified above x = 0, 1, ∞, but its ramification structure is 1 + 4 + 1 = (2)3 = (3)2.
Taking the multiplicities in this ramification structure into account, one finds that if the 2 F 1 of Definition 3.1 is lifted along x = R(u) rather than x =R(s), the P-symbol identity (6.3) is replaced by This P-symbol has five singular points (at most; fewer if n = 0 or m = 0). By the preceding explanation, the points u = 0, 1, ∞ represent 1, 4, 1 vertices of the octahedron, and each of u = −(2 ± √ 3) 2 represents a cycle of four face-centers.
, the P-symbol of which appears in Eq. (6.5b). The function r m n is the Frobenius solution associated to the zero characteristic exponent of the singular point u = 0. It satisfies eight differential recurrences of the form in which ∆(n, m) = (0, ±1), (±1, 0) and (±1, ±1). For each recurrence, the exponents σv, σe, σ f , the exponents εv, εe, ε f , and the prefactor K are listed in Table 1.
Proof The differential equation comes by applying to the appropriate Gauss equation of the form (6.1) the changes of variable that perform (i) the lifting along u → x = R(u), and (ii) the multiplication by [pe(u)]  (27)].) And if ∆(n, m) = (±1, ±1), the 2 F 1 (R(u)) in the definition of r m n (u) has its parameters shifted by integers. (See Definition 3.1.) By some calculus, one can change the independent variable in the relevant differential recurrences of Jacobi from x = R(u) to u, thereby obtaining the final four recurrences in Table 1 (the diagonal ones). The change uses the fact that u 3/4 dR/du equals −108 p 3/4 v p 2 f /p 3 e , and the details are straightforward.
The first four recurrences in the table, with ∆(n, m) = (0, ±1), (±1, 0), come a bit less easily, because they shift the 2 F 1 parameters in Definition 3.1 by halfintegers rather than integers. But by examination, Definition 3.1 is equivalent to Table 1 Parameters for the differential recurrences of Theorem 6.1.
This follows by a quadratic hypergeometric transformation, u being related to t by u = −3t 2 , and R to S by R = S 2 /(S − 2) 2 . When ∆(n, m) = (0, ±1) or (±1, 0), the parameters of the 2 F 1 in (6.6) are shifted by integers, and the same technique can be applied.

⊓ ⊔
The four three-term non-differential recurrences in Theorem 3.3 follow by a familiar elimination procedure from the differential recurrences of Theorem 6.1, taken in pairs. They are analogous to the contiguity relations (or 'contiguous function relations') of Gauss, for 2 F 1 , which follow by elimination from the differential recurrences of Jacobi, though Gauss did not derive them in this way.
The explicit formulas for the functions r m n with small n, m given in § 3.1 (see Eqs. (3.3),(3.4)) also follow from the differential recurrences of Theorem 6.1.  Thus when m ≥ 0, r m 0 is a degree-2m hypergeometric polynomial.
Moreover, for any n, m ≥ 0, r m n is a polynomial of degree 3n + 2m.
Proof When n = 0, L m n f = 0 loses two singular points and degenerates to a Gauss hypergeometric equation of the form (6.1), with independent variable u and parameters a = −2m, b = − 1 4 − 3m, c = 3 4 − m. Hence r m 0 (u) has the claimed representation, and if m ≥ 0, is a degree-2m polynomial in u. It follows by induction from the differential recurrence with ∆(n, m) = (+1, 0) that r m n (u) is a polynomial in u for all n ≥ 0. It must be of degree 3n + 2m, because in the P-symbol of L m n [see (6.5b)], the only characteristic exponent at u = ∞ that is a (nonpositive) integer is −2m − 3n.

⊓ ⊔
The statement of this theorem includes additional claims that were made in § 3.1. The following related theorem mentions the Heun function Hn(a, q; α, β, γ, δ | z), for the definition of which see [34]. This is a Frobenius solution (at z = 0) of a canonical Fuchsian differential equation that has four singular points, namely z = 0, 1, a, ∞, and an 'accessory' parameter q that unlike α, β, γ, δ, does not affect their characteristic exponents. It has a convergent expansion ∞ k=0 h k z k , where the {h k } ∞ k=0 satisfy a second-order recurrence with coefficients quadratic in k.
Proof If m = 0, the u = 1 singular point of L m n drops out, i.e., becomes ordinary, and The substitution z = −(2 + √ 3) 2 u reduces L 0 n f = 0 to the standard Heun equation [34], with the stated values of a, q; α, β, γ, δ. The recurrence (6.8), based on a second-order difference operator, comes by substituting f = r 0 n = ∞ k=0 a k u k into L 0 n f = 0.

⊓ ⊔
For general (n, m) ∈ Z 2 , L m n f = 0 has five singular points. The theory of such generalized Heun equations is underdeveloped at present, but the coefficients of their series solutions are known to satisfy third-order (i.e., four-term) recurrences.
Proof The recurrence comes by substituting f = r m n = ∞ k=0 a k u k into L m n f = 0.

⊓ ⊔
It can be shown that if m = 0, the third-order (i.e. generalized Heun) difference operator in (6.9) has the second-order (i.e. Heun) difference operator in (6.8) as a right factor; and if n = 0, it has a first-order (i.e. hypergeometric) difference operator as a right factor, which is why the representation in Theorem 6.2 exists. The coefficients of all these difference operators are quadratic in k.
As stated in Theorem 3.2, it is not merely the case that when n, m ≥ 0, the rational function r m n = r m n (u) is a polynomial of degree 3n + 2m. In each quadrant of the (n, m)-plane, it is the quotient of a polynomial of known degree (the numerator) by a known polynomial (the denominator). To obtain the formulas in Theorem 3.2 that refer to quadrants other than the first, reason as follows. Consider the second formula: it says that if n, m ≥ 0, r −m−1 n (u) equals a polynomial of degree 1 + 3n + 2m, divided by (1 − u) 3+4m . This is proved by induction on n, the base case (n = 0) being which comes from (6.7) by Euler's transformation of 2 F 1 . The inductive step uses the differential recurrence with ∆(n, m) = (+1, 0), as in the proof of Theorem 6.2.
In the same way, the third and fourth formulas follow from the ∆(n, m) = (−1, 0) recurrence.
One sees from the four formulas in Theorem 3.2 that irrespective of quadrant, r m n ∼ const × u 3n+2m , which partially confirms the claims of Theorem 3.1(i,ii). A consequence of this asymptotic behavior is that besides being the Frobenius solution associated to the exponent 0 at u = 0, r m n is the Frobenius solution associated to the exponent −2m − 3n at u = ∞, which appeared in the P-symbol (6.5b).
Theorem 3.1 states specifically that r m n ∼ d m n × u 3n+2m , with d m n defined in (3.2). This too is proved by induction. The base case (n = 0) has sub-cases m ≥ 0 and m ≤ 0, which follow by elementary manipulations from (6.7) and (6.10), respectively. The inductions toward n ≥ 0 and n ≤ 0 come from the differential recurrences with ∆(n, m) = (±1, 0), the u → ∞ asymptotics of which yield expressions for d m n±1 /d m n . As one can check, these two expression agree with what (3.2) predicts.
The only claim in § 3.1 remaining to be proved is Theorem 3.1(iii): the statement that the conjugate function r m n = r m n (u) is related to r m n = r m n (u) by where the operatorL m n was defined in (6.4). By inspection, it has the simpler representatioñ Because the coefficient function q f /q 2 v diverges at the endpoints s = 0, 1, this Sturm-Liouville problem is typically a singular one. To avoid a discussion of endpoint classifications and boundary conditions, it is best to derive orthogonality results not fromL m n , but rather from the Love-Hunter biorthogonality relation (1.1), i.e., which holds if µ ∈ (−1, 1) and ν, ν ′ differ by a nonzero even integer. (See [21, Appendix] for a proof.) Equation (7.2) is a relation of orthogonality between the eigenfunctions of a singular boundary value problem based on (2.1), the associated Legendre equation (i.e., P µ ν0+2n (z), n ∈ Z), and the eigenfunctions of the adjoint boundary value problem (i.e., P −µ ν0+2n (−z), n ∈ Z). The first problem is non-selfadjoint because the boundary conditions that single out P µ ν0+2n (z), n ∈ Z, as eigenfunctions are not self-adjoint.
However, one feature of the operatorL m n must be mentioned. Iff =f(s) solvesL m nf = 0, then so does (1−s) 1+12m+12nf ((1 + s)/(1 − s)). This claim can be verified by a lengthy computation, but its correctness is indicated by the P-symbol ofL m n , which appeared in (6.3c). The map s → (1 + s)/(1 − s) is a 90 • rotation of the s-sphere, and hence of the inscribed octahedron, around the axis through the equatorial vertices s = ±i. This rotation takes vertices to vertices, edges to edges, and faces to faces. The subsequent multiplication by (1 − s) 1+12m+12n shifts the characteristic exponents at the most affected vertices (s = 1, ∞) to the values they had before the rotation.  (cos θ) in Theorem 3.5 into (7.2), and change the variable of integration from z = cos θ to u = B − /B + , and then to s = u 1/4 . The involution z → −z corresponds to s → (1 − s)/(1 + s).

⊓ ⊔
This biorthogonality theorem is formulated so as to indicate its close connection to Sturm-Liouville theory: evaluating the integral over 0 < s < 1 computes the inner product of the two square-bracketed factors in the integrand, which come from P µ ν (z) and P −µ ν ′ (−z), with respect to the weight function q 2 v /q 2 f . The two factors are eigenfunctions of adjoint Sturm-Liouville problems on 0 < s < 1 (i.e., ones with adjoint boundary conditions), with different eigenvalues.
Theorem 7.1 cannot be extended to general m ∈ Z, because the integral diverges unless m = 0 or m = −1, owing to rapid growth of one or the other of the bracketed factors at each of the endpoints s = 0, 1. This divergence follows readily from the results on r m n given in Theorems 3.1 and 3.2. Alternatively, the divergence arises from the Ferrers function P µ ν not lying in L 2 [−1, 1] when µ is non-integral, unless Re µ ∈ (−1, 1).
The formulas for the tetrahedral Ferrers functions P ±( 1 3 +n) − 3 4 −m given in Theorem 4.3 (cases m = 0, −1) can also be substituted usefully into the Love-Hunter relation (7.2). But the resulting statement of biorthogonality is more complicated than Theorem 7.1 and is not given here.

Cyclic and Dihedral Formulas (Schwarz Classes O and I)
This section derives parametric formulas for Legendre and Ferrers functions that are cyclic or dihedral. The formulas involve the Jacobi polynomials P (α,β) n and are unrelated to the octahedral and tetrahedral ones in § § 3, 4, and 5. They are of independent interest, and subsume formulas that have previously appeared in the literature.
As used here, 'cyclic' and 'dihedral' have extended meanings. The terms arise as follows. The associated Legendre equation (2.1) has (µ, µ, 2ν +1) as its (unordered, unsigned) triple of characteristic exponent differences. By the results of Schwarz on the algebraicity of hypergeometric functions, this differential equation will have only algebraic solutions if (ν+ 1 2 , µ) lies in (± 1 2 , ± 1 2k )+Z 2 or (± 1 2k , ± 1 2 )+Z 2 , for some positive integer k. These restrictions cause the equation to lie in Schwarz's cyclic class (labelled O here), resp. his dihedral class I. The terms refer to the projective monodromy group of the equation, which is a (finite) subgroup of P SL(2, R).
However, the formulas derived below are more general, in that they allow k to be arbitrary: they are formulas for continuously parametrized families of Legendre and Ferrers functions, which are generically transcendental rather than algebraic. Because of this, we call a Legendre or Ferrers function cyclic, resp. dihedral, if (ν + 1 2 , µ) lies in (± 1 2 , * )+Z 2 , resp. ( * , ± 1 2 )+Z 2 ; the asterisk denoting an unspecified value. That is, the degree ν should be an integer or the order µ a half-odd-integer, respectively.
Explicit formulas in terms of Jacobi polynomials are derived in § 8.1, and how dihedral Ferrers functions can be used for expansion purposes is explained in § 8.2.

Explicit Formulas
The Jacobi polynomials P (α,β) n (z) are well known [11, § 10.8]. They have the hypergeometric and Rodrigues representations and are orthogonal on [−1, 1] with respect to the weight function (1 − x) α (1 + x) β , if α, β > −1 and the weight function is integrable. Legendre and Ferrers functions that are cyclic (i.e., of integer degree) are easily expressed in terms of Jacobi polynomials.
hold when n is a non-negative integer, for z ∈ (1, ∞) and ξ ∈ (0, ∞). (In the degenerate case when µ − n is a positive integer, P µ hold when n is a non-negative integer, for z ∈ (−1, 1), ξ ∈ (−∞, ∞), and θ ∈ (0, π). (In the degenerate case when µ − n is a positive integer, P µ respectively. However, the coefficients in (2.6) and (2.7) diverge when µ ∈ Z. Hence, following this approach to formulas forQ m 2 ) , when n is a nonnegative integer and m an integer, requires the taking of a limit. In the commonly encountered case when −n ≤ m ≤ n (but not otherwise), the resulting expressions turn out to be logarithmic. Such expressions can be computed in other ways [11, § 3.6.1]. Perhaps the best method is to expressQ m − 1 2 ±(n+ 1 2 ) in terms of a 2 F 1 by using (2.8d), and then use known formulas for logarithmic 2 F 1 's [5].
Legendre and Ferrers functions that are dihedral (i.e., are of half-odd-integer order) are the subject of the following theorems. For conciseness, a special notation is used: [A|B] ± signifies A, resp. B, in the +, resp. − case; and {C} α,± , where C depends on α, signifies the even or odd part of C under α → −α, i.e., hold when m is a non-negative integer, for z ∈ (1, ∞) and ξ ∈ (0, ∞).

⊓ ⊔
In these formulas, the proportionality ofQ hold when m is a non-negative integer, for z ∈ (1, ∞) and ξ ∈ (0, ∞); it being understood in the 'minus' case that when α = −m, . . . , m and there is an apparent division by zero, each right-hand side requires the taking of a limit.
Proof Combine the P →Q reduction (2.5) with the results in Theorem 8.3.
hold when m is a non-negative integer, for θ ∈ (0, π). In the sub-cases α = −m, . . . , m of the 'minus' case, the apparent division by zero in the first formula is handled by interpreting its right-hand side in a limiting sense; but the division by zero in the second formula causes both its sides to be undefined. Proof The first formula follows by analytic continuation of the latter formula in Theorem 8.4; in effect, by letting ξ = −iθ. The second formula then follows from the Q → P reduction (2.7), after some algebraic manipulations.

Dihedral Ferrers Functions and Love-Hunter Expansions
In this subsection, we show that an expansion in dihedral Ferrers functions can be, in effect, an expansion in Chebyshev polynomials (of the fourth kind); and as an application, show that the result of [32] on the convergence of Love-Hunter expansions can be slightly extended.
The These hold for θ ∈ (0, π), the α = 0 case of the former requiring the taking of a limit.
In order (i) to make endpoint behavior more symmetrical and less divergent, and (ii) to study endpoint convergence, Pinsky [32] has proposed modifying Love-Hunter expansions by treating [(1 − z)/(1 + z)] µ/2 P µ ν (z) rather than P µ ν (z) as the expansion function. By (2.8a), this amounts to replacing each P µ ν (z) by the 2 F 1 function in terms of which it is defined; i.e., performing a hypergeometric expansion.
By standard Fourier series theory, the expansion of g = g(u) in the T k (u), when g is piecewise continuous on −1 ≤ u ≤ 1, will converge to g at all points of continuity, and in general to [g(u+) + g(u−)] /2. But (see [24, § 5.8.2]), if one writes z = 1−2u 2 (so that u = sin(θ/2) if z = cos θ), then W j (z) equals (−1) j u −1 T 2j+1 (u). Therefore an expansion of f = f (z) in the fourth-kind W j (z) on −1 ≤ z ≤ 1 is effectively an expansion of g(u) = uf (1 − 2u 2 ) on −1 ≤ u ≤ 1 in the first-kind T k (u), each even-k term of which must vanish. The theorem follows.

⊓ ⊔
It is useful to compare this convergence result, which refers to an expansion of f in the Ferrers functions P − 1 2 2n , with the pointwise convergence result of [32]. The latter deals with an expansion in the functions P µ ν0+2n , where ν 0 is arbitrary and µ ∈ (− 1 2 , 1 2 ). However, it requires that f be piecewise smooth, not merely piecewise continuous.

Ladder Operators, Lie Algebras, and Representations
In the preceding sections, explicit formulas for the Legendre and Ferrers functions in the octahedral, tetrahedral, dihedral, and cyclic families were derived. Each such family (in the first-kind Ferrers case) is of the form {P µ0+m ν0+n (z = cos θ)}, where ν 0 , µ 0 are or may be fractional, and (n, m) ranges over Z 2 . In this section, the connection between such a family and conventional SO(3)-based harmonic analysis on the sphere S 2 = SO(3)/SO(2), coordinatized by the angles (θ, ϕ), is briefly explored.
The connection goes through the corresponding family of generalized spherical harmonics, P µ ν (cos θ)e iµϕ , with (ν, µ) ∈ (ν 0 , µ 0 ) + Z 2 . But the connection is not as strong as one would like. If ν 0 , µ 0 are rational but not integral, these harmonic functions will not be single-valued on the symmetric space S 2 . (In the cases of interest here, each P µ ν (z) in the family is algebraic in z, and they can be viewed as finite-valued.) They may not be square-integrable, because the leading behavior of P µ ν (z) as z → 1 − is proportional to (1 − z) −µ/2 unless µ is a positive integer. For these reasons, the focus is on the action of Lie algebras (of 'infinitesimal transformations') on a function family of this type, specified by (ν 0 , µ 0 ), rather than the action of a Lie group such as SO(3). The space spanned by the classical spherical harmonics Y m n (θ, ϕ) ∝ P m n (cos θ)e imϕ , with n ≥ 0 and m ∈ Z, admits an action of the rotation group SO(3). The Lie algebra so(3, R) of 3 × 3 real skewsymmetric matrices can be represented by differential operators on S 2 , with real coefficients, and acts on the space of spherical harmonics. The resulting infinitedimensional representation is reducible: for n = 0, 1, 2, . . . , it includes the usual (2n + 1)-dimensional representation on the span of Y −n n , . . . , Y n n . But, so(3, R) is not the only Lie algebra to be considered.
A larger Lie algebra than so(3, R) acts naturally on the spherical harmonics, or rather, on the (regular) solid harmonics r n Y m n (θ, ϕ), which satisfy Laplace's equation on R 3 . (See [28, § 3.6].) This is the 10-dimensional real Lie algebra so(4, 1) that is generated by 'ladder' operators that increment and decrement the degree n, as well as the order m. They are represented by differential operators on R 3 , with real coefficients. The real span of these operators exponentiates to the Lie group SO 0 (4, 1), which contains as subgroups (i) the 3-parameter group SO(3) of rotations about the origin, (ii) a 3-parameter abelian group of translations of R 3 , (iii) a 1-parameter group of dilatations (linear scalings of R 3 ), and (iv) a 3-parameter abelian group of 'special conformal transformations.' The last are quadratic rational self-maps of R 3 (or rather the real projective space RP 3 , because they can interchange finite and infinite points).
The preceding results, now standard, are extended below to any family of generalized solid harmonics {r ν P µ ν (cos θ)e iµϕ }, with (ν, µ) ∈ (ν 0 , µ 0 ) + Z 2 for specified ν 0 , µ 0 . In § 9.1, the differential and non-differential recurrences on ν and µ are derived. (See Theorems 9.1 and 9.2.) In § 9.2, it is shown that the ladder operators in the differential recurrences generate a 10-dimensional real Lie algebra, and an isomorphism from this algebra not to so(4, 1) but to so(3, 2) is exhibited. The treatment closely follows Celeghini and del Olmo [4], but the explicit isomorphism in Theorem 9.3 is new.
In § 9.3, it is shown that irrespective of (ν 0 , µ 0 ), the representation of so(3, 2) [or of so(4, 1) or so(5, R)] carried by the solid harmonics r ν P µ ν (cos θ)e iµϕ with (ν, µ) ∈ (ν 0 , µ 0 ) + Z 2 is of a special type: its quadratic Casimir operator takes a fixed value, and its quartic one vanishes. (See Theorem 9.6.) The former fact was found in [4], but the latter is new. The representation of so(3, 2) on the solid harmonics of integer degree and order, and its representation on the ones of halfodd-integer degree and order, have irreducible constituents that are identified as the known Dirac singleton representations of so(3, 2).

Differential and Non-differential Recurrences
In any family {P µ0+m ν0+n (z)} (n,m)∈Z 2 , where P can be taken as any of P, Q, P,Q, any three distinct members are linearly dependent, over the field of functions that are rational in z and √ 1 − z 2 (Ferrers case) or √ z 2 − 1 (Legendre case). In particular, any three contiguous members are so related, by a three-term ladder recurrence.
The underlying recurrences are differential ones, which generally permit any single P µ ν and its derivative to generate any member contiguous to it, as a linear combination; and by iteration, to generate any P µ+∆µ ν+∆ν in which ∆(ν, µ) ∈ Z 2 .
The Legendre functions P µ ν , Q µ ν [the latter unnormalized, i.e., the functions e µπiQµ ν ] satisfy recurrences obtained from the preceding by (i) multiplying the right-hand side by a sign factor, equal to i ε1+∆µ ; and (ii) replacing 1 − z 2 by z 2 − 1.

⊓ ⊔
It was noted in § 2 that if ν + µ is a negative integer, Q µ ν and Q µ ν are generally undefined (though there are exceptions). The recurrences for Q µ ν and Q µ ν in Theorems 9.1 and 9.2 remain valid in a limiting sense even when (ν, µ) is such that one or more of the functions involved is undefined.
More concretely, so(p, q) can be realized as the real span of the n × n matrices M ab , 1 ≤ a < b ≤ n, where M ab = Γ E ab −E ba Γ . In this, E ab is the n×n matrix with a 1 in row a, column b, and zeroes elsewhere. One often extends the size-n 2 basis {M ab } to a 'tensor operator,' i.e., a skew-symmetric n×n matrix of elements (M ab ), by requiring that M ba = −M ab for 1 ≤ a, b ≤ n. The commutation relations [M ab , M cd ] = g ad M bc + g bc M ad − gacM bd − g bd Mac (9.11) are easily checked.
Theorem 9.3 The real Lie algebra generated by J ± , K ± is isomorphic to so(3, 2), an isomorphism being specified by the tensor operator are the 'compact' and 'non-compact' subspaces. (The terms refer to the Lie subgroups of SO 0 (3, 2) to which they exponentiate.) Real Lie algebras isomorphic to so(4, 1) and so(5, R) can be obtained by Weyl's trick of redefining some or all of the basis elements of p to include 'i' factors. In doing this, a slightly changed notation will be useful. For X = J, P, C, P C + , P C − , define the 'Cartesian' elements so that X ± = X 1 ± iX 2 and (X 1 , X 2 , X 3 ) = (X 1 , iX 2 , X 3 ).
Theorem 9.4 (i) The real span of iJ 1 , iJ 2 , iJ 3 ; P 1 , P 2 , P 3 ; C 1 , C 2 , C 3 ; D, or equivalently of iJ 1 , iJ 2 , iJ 3 ; P C ± 1 , P C ± 2 , P C ± 3 ; D, is a real Lie algebra isomorphic to so(4, 1), an isomorphism being specified by the tensor operator iD is a real Lie algebra isomorphic to so (3,2), an isomorphism being specified by the tensor oper- Proof (i) Multiply the second row and the second column of the (M ab ) in Theorem 9.3 by 'i', and (innocuously) interchange the second and third rows, and the second and third columns. (ii) Continuing (or in a sense reversing), multiply the last row and the last column by 'i'.
⊓ ⊔ Theorem 9.5 The real span of iJ 1 , iJ 2 , iJ 3 ; P C + 1 , P C + 2 , P C + iD is a real Lie algebra isomorphic to so(5, R), an isomorphism being specified by the tensor operator Proof Multiply the first row and the first column of the (M ab ) in part (i) of Theorem 9.4 by 'i'.

⊓ ⊔
With 'i' factors in basis elements, the so(3, 2), so(4, 1) and so(5, R) of Theorems 9.4 and 9.5 look awkward. But it follows from (9.3), (9.5), and (9.12) that each basis element in the so(4, 1) of Theorem 9.4(i), i.e., each of iJ i , P i , C i , and D, is realized by a differential operator in r, θ, ϕ with real coefficients. This is not the case for the basis elements of so(3, 2) and so(5, R).
The geometric significance of this realization of so(4, 1) is revealed by changing from spherical coordinates (r, θ, ϕ) to Cartesian ones, (x 1 , x 2 , x 3 ) on R 3 . Using x := (x 1 , x 2 , x 3 ) = (r sin θ cos ϕ, r sin θ sin ϕ, r cos θ), one finds that where ∂ i := d/dx i and i, j, k is a cyclic permutation of 1, 2, 3. That is, the iJ i generate rotations about the origin, the P i generate translations, and D generates dilatations (linear scalings of R 3 ). The C i generate special conformal transformations, which are degree-2 rational maps of R 3 (or rather RP 3 ) to itself. The commutation relations written in terms of P C ± i = 1 2 (P i ± C i ), follow either from (9.9),(9.10), from (9.11), or from (9.13). Here, the summation convention of tensor analysis is employed. The Levi-Cività tensor ǫ ijk is skew-symmetric in all indices, with ǫ 123 = +1, and δ ij is the Kronecker delta. Together with the formulas 15c) express all these differential operators in terms of the original J ± , K ± , R ± , S ± ; J 3 , K 3 of (9.3),(9.5). The ten operators in (9.13) span (over R) the Lie algebra of conformal differential operators on R 3 , which is known to have an so(4, 1) structure. (See Miller [28, § 3.6].) This is the symmetry algebra of the Laplacian ∇ 2 on R 3 , which comprises all real first-order operators L for which [L, ∇ 2 ] ∝ ∇ 2 , i.e., for which [L, ∇ 2 ] has ∇ 2 as a right factor. It can be viewed as acting on any suitable space of functions on R 3 , and exponentiates to the group SO 0 (4, 1) of conformal transformations, realized as flows on R 3 (or RP 3 ). But the starting point used here was their action on the span of the generalized solid harmonics S µ0+m ν0+n with (n, m) ∈ Z 2 , which are (multi-valued) solutions of Laplace's equation.
In the physics literature on conformal Lie algebras and groups, the terms 'x i ' in (9.13c) and ' 1 2 ' in (9.13d) often appear as 2δx i and δ respectively, where δ is the so-called scaling dimension; though the resulting commutation relations do not involve δ. The value δ = 1 2 is specific to the symmetry algebra of the Laplacian. There are many variations on the present technique of using differential recurrences to construct real Lie algebras, realized by differential operators, that are isomorphic to the real forms of so(5, C). The solid harmonics S µ ν that were employed here are extensions to R 3 of the (surface) spherical harmonics P µ ν (cos θ)e iµϕ on the symmetric space S 2 = SO(3)/SO(2). If not Ferrers but Legendre functions were used, the starting point would be the hyperboloidal ones P µ ν (cosh ξ)e iµϕ , defined using coordinates (ξ, ϕ) on the hyperboloid H 2 = SO(2, 1)/SO(2), i.e., the surface 3 + const = 0. Their extensions to R 3 satisfy the (2 + 1)-dimensional wave equation, rather than Laplace's equation. (See [28,Chap. 4] and [9].) But isomorphic algebras could be constructed.
These infinite-dimensional representations are restrictions of the representation of the common complexification so(5, C), which is carried by the (complex) span of the family. They are generically irreducible, and are also generically non-skew-Hermitian, so that except in special cases, they do not exponentiate to unitary representations of the corresponding Lie groups, even formally. This will now be investigated.
Each of the three real Lie algebras is of rank 2, so the center of its universal enveloping algebra is generated by two elements, called Casimir invariants; and any irreducible representation must represent each Casimir by a constant. The analysis of such representations resembles the unified classification of the irreducible representations of so(2, 1) and so(3, R), the real forms of so(3, C), which is well known. (See, e.g., [8,Chap. 3].) In this, representations are classified by the value taken by their (single) Casimir, and by their reductions with respect to a (1-dimensional) Cartan subalgebra. This leads to an understanding of which representations are skew-Hermitian and which are finite-dimensional. However, no comparable unified approach to all representations of so(3, 2), so(4, 1), and so(5, R) seems to have been published. The literature has dealt almost exclusively with the skew-Hermitian ones. (so (3,2) and so(4, 1) are treated separately in [10,12] and [6], and so(4, 1) and so(5, R) are treated together in [18].) The starting point is the complexification so(5, C), which is generated over C by J ± and K ± , the ladder operators on the order and degree. It is the complex span of J ± , K ± , R ± , S ± ; J 3 , K 3 , each of which is represented as in (9.4) and (9.6) by an infinite matrix indexed by (ν, µ) ∈ (ν 0 , µ 0 )+Z 2 . The elements J 3 , K 3 span a Cartan subalgebra (an abelian subalgebra of maximal [complex] dimension, here 2), which is represented diagonally: When the representation of so(5, C) is reduced with respect to this subalgebra, it splits into an infinite direct sum of 1-dimensional representations, indexed by (ν, µ). The corresponding real Cartan subalgebras of the so(3, 2) and so(4, 1) in Theorems 9.3 and 9.4(i) are the real spans of {J 3 , K 3 } and {iJ 3 , K 3 }. For the so(3, 2) and so(5, R) in Theorems 9.4(ii) and 9.5, they are the real span of {iJ 3 , iK 3 }. (Recall that D := K 3 .) Only for the last two will the real Cartan subalgebra be represented by skew-Hermitian matrices; in fact, by imaginary diagonal ones. It is readily verified that J ± , K ± , R ± , S ± ; J 3 , K 3 can serve as a Cartan-Weyl basis of so(5, C), their complex span. That is, when the adjoint actions of H 1 := J 3 and H 2 := K 3 on this 10-dimensional Lie algebra are simultaneously diagonalized, the common eigenvectors ('root vectors') include J ± , K ± , R ± , S ± . The associated roots α ∈ R 2 are 2-tuples of eigenvalues, which can be identified with the displacements ∆(ν, µ), i.e., ±(0, 1), ±(1, 0), ±(1, 1), ±(1, −1). These form the B 2 root system. One can write where Eα is the root vector associated to root α. The commutators [Eα, E β ] also prove to be consistent with the B 2 root system. The Casimir invariants of so(5, C) and its three real forms can be computed from the commutation relations of the Cartan-Weyl basis elements. (For instance, the Killing form for the algebra yields a quadratic Casimir.) But it is easier to express them using the tensor operator M ab of any of Theorems 9.3, 9.4, and 9.5. As elements of the universal enveloping algebra, the two Casimirs, quadratic and quartic, are defined thus [10,12]: where w a = 1 8 ǫ abcde M bc M de and the summation convention is employed, indices being raised and lowered by the tensors Γ −1 = (g ab ) and Γ = (g ab ). The Levi-Cività tensor ǫ abcde is skew-symmetric in all indices, with ǫ 12345 = +1. The normalization and sign conventions are somewhat arbitrary. Theorem 9.6 In the representation of the universal enveloping algebra of any of the real Lie algebras so(3, 2), so(4, 1), and so(5, R) on the span of the generalized solid harmonics S µ0+m ν0+n (r, θ, ϕ), (n, m) ∈ Z 2 , the Casimirs c 2 and c 4 are represented by the constants − 5 Proof By the expressions for M ab , Γ given in any of Theorems 9.3, 9.4, and 9.5, c 2 = J · J − P C + · P C + + P C − · P C − + D 2 (9.16a) where {·, ·} is the anti-commutator. This expresses c 2 in terms of J 3 , K 3 and the root vectors. The formula (9.16b) can be viewed as subsuming J 2 3 + 1 which is the Casimir of the so(3, R) subalgebra spanned by {J + , J − , J 3 }; and K 2 3 − 1 2 {K + , K − }, which is the Casimir of the so(2, 1) subalgebra spanned by {K + , K − , K 3 }; and also, the Casimirs of the remaining two so(2, 1) subalgebras. From the representations (9.4),(9.6) of J ± , K ± , R ± , S ± and J 3 , K 3 as infinite matrices, one calculates from (9.16b) that c 2 (like J 3 , K 3 ) is diagonal in (n, m), with each diagonal element equaling − 5 4 . For so(5, R), which is representative, the five components of w a include (i) the scalar i J · P C + , (ii) the three components of the vector −i P C − × P C + + DJ, and (iii) the scalar J · P C − . These expressions, involving the scalar and vector product of three-vectors, must be interpreted with care: any product AB of two Lie algebra elements signifies the symmetrized product 1 2 {A, B}. But by direct computation, one finds from (9.7),(9.8) and the infinite matrix representations (9.4),(9.6) that each component of w a is represented by the zero matrix, even (surprisingly) without symmetrization.

⊓ ⊔
This result is plausible, if not expected. In any unitary representation of a semi-simple Lie group G on L 2 (S), S being a homogeneous space G/K of rank 1, all Casimir operators except the quadratic one must vanish. (See [3] and [15, Chap. X].) Admittedly, the present representations of so(4, 1), by real differential operators acting on multi-valued, non-square-integrable functions, are non-skew-Hermitian, and cannot be exponentiated to unitary representations of SO 0 (4, 1) of this 'most degenerate' type. The value − 5 4 computed for the quadratic Casimir c 2 , irrespective of (ν 0 , µ 0 ), can be viewed as the value of j(j + 1), where j is a formal 'angular momentum' parameter equal to − 1 2 ± i. For each (ν 0 , µ 0 ), the resulting representation of the real Lie algebra g = so(3, 2), so(4, 1) or so(5, R), or its universal enveloping algebra U(g), on the span of the generalized solid harmonics S µ0+m ν0+n , (n, m) ∈ Z 2 , can be viewed linearalgebraically: as a homomorphism ρ of real vector spaces, taking g (or U(g)) into the space of infinite matrices indexed by (n, m). For each basis element A ∈ g, ρ(A) is determined by (9.4),(9.6); and because the basis elements given in Theorems 9.4 and 9.5 include 'i' factors, the matrix elements of ρ(A) may be complex.