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 nonclassical 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)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {so}(5,\mathbb {C})$$\end{document} is generated by the ladder operators that shift the degree and order of the corresponding solid harmonics. All such representations of so(5,C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {so}(5,\mathbb {C})$$\end{document} are shown to have a common value for each of its two Casimir invariants. The Dirac singleton representations of so(3,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {so}(3,2)$$\end{document} are included.

It is not usually the case that P μ ν (z) and P μ ν (z) are elementary functions, unless of course ν and μ are integers. This may be why such expansions as (1.2) have been used infrequently. In this paper, we derive explicit, trigonometrically parametrized formulas for several families of Legendre functions, expressing P where (n, m) ∈ Z 2 . Here, r m n = r m n (u) is an 'octahedral' rational function that if n, m ≥ 0 is a polynomial of degree 3n +2m in u; in the base case n = m = 0, it equals unity. It satisfies differential recurrences on n and m, and three-term nondifferential 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 secondorder 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,Sect. 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) = (− 1 6 , 1 4 ) + (n, m), as in (1.3), are ( 1 4 , 1 4 , 2 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 μ 0 +m ν 0 +n (z = cos θ)}, (n, m) ∈ Z 2 , and the corresponding spherical harmonics {Y μ 0 +m ν 0 +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 μ 0 +m ν 0 +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 μ 0 +m ν 0 +n (z)} or {Y μ 0 +m ν 0 +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 infinite-dimensional representation of the Lie algebra so(3, R) on the span of {Y μ 0 +m ν 0 (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 −ν 0 ν 0 , . . . , Y ν 0 ν 0 .) There is also a representation of so(2, 1) on the span of {Y μ 0 ν 0 +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 10dimensional, rank-2 complex Lie algebra isomorphic to so(5, C), which for any (ν 0 , μ 0 ), acts differentially on the family {r ν 0 +n Y μ 0 +m ν 0 +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 Δ(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 nondifferential 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, and 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 ν 0 +n Y μ 0 +m ν 0 +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 Sect. 2, facts on Legendre/Ferrers functions that will be needed are reviewed. In Sect. 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 Sects. 4 and 5. In Sect. 6, the results in Sect. 3 are proved. In Sect. 7, Love-Hunter biorthogonality is related to Sturm-Liouville biorthogonality. In Sect. 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 Sect. 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. At fixed real μ, Eq. (2.1) can be viewed as a singular Sturm-Liouville equation on the real Ferrers domain [−1, 1], the endpoints of which are of Weyl's 'limit circle' type if μ ∈ (−1, 1). (See [13].) In this case, all solutions p = p(z) lie in L 2 [−1, 1], irrespective of ν, but the same is not true when μ / ∈ (−1, 1), which is why such orthogonality relations as (1.1) can only be obtained if μ ∈ (−1, 1), or more generally if Re μ ∈ (−1, 1).
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 with the exponent −μ/2 at z = 1, and the second Legendre function Q μ ν is associated with 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 typically involves replacing a factor (z − 1) −μ/2 by (1 − z) −μ/2 ; for instance, P −1 1 (z), Also, owing to (2.3b), Q μ ν is undefined if and only ifQ μ ν is. Equation (2.1) is invariant under ν → −ν − 1, μ → −μ, and z → −z, so that in nondegenerate cases, the Legendre and Ferrers functions with ν replaced by −ν − 1, μ by −μ, and/or z by −z, can be expressed as combinations of any two (at most) of P μ ν ,Q μ ν , P μ ν , Q μ ν . Some 'connection' formulas of this type, which will be needed below, are P (2.6) and the Q → P reduction (See [11].) It follows from ( (2.8d) (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 , the denominator parameter c is a nonpositive 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 Sects. 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 Sect. 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, [For later use, note that (3.1) is familiar from trigonometry as a 'triple-angle' formula: 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 4 , 1 3 , 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. The functional form of the octahedral functions that are not polynomials, which are indexed by (n, m) ∈ Z 2 \ Z 2 ≥ , is not complicated. Theorem 3.2 For any n, m ≥ 0, 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 three-term nondifferential recurrences. The former are given in Sect. 6 (see Theorem 6.1), and the latter are as follows.

Theorem 3.3
The octahedral functions r m n = r m n (u), indexed by (n, m) ∈ Z 2 , satisfy second-order (i.e., three-term) recurrences on m and n; namely, where p v , p e , p f are the polynomials in u, satisfying p 2 e − p 3 f + 108 p v = 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, These also follow from the recurrences of Theorem 3.3. The recurrences are nonclassical, not least because they are bilateral: they extend to n, m < 0. It is shown in Sect. 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 Sect. 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 , both lying in Schwarz's octahedral class V. An interesting application of these formulas to the evaluation of certain Mehler-Dirichlet integrals appears in Theorem 3.6.
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 Moreover, and more fundamentally, 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).

Theorem 3.4 The formulas
[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 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 .
In the following, the circular-trigonometric functions B ± , positive on (0, π), are defined by

Theorem 3.5 The formulas
[Note that as θ increases from 0 to π , the argument u = B − /B + of the first r m n , which satisfies T (u) = − tan 2 (θ/3) and R(u) = − tan 2 θ , increases from 0 to 1.] Proof The proof is accomplished by analytic continuation of Theorem 3.4, or in effect, by letting ξ = iθ .

Tetrahedral Formulas (Schwarz Class II)
The following theorem gives trigonometrically parametrized formulas for the second Legendre functionQ

Theorem 4.1 The formulas
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]. . Moreover, it leads to the following two theorems.
Proof Combine the P →Q reduction (2.5) with the results in Theorem 4.1.
In Theorem 4.3, the hyperbolic-trigonometric functions C ± , positive on (−∞, ∞), are defined by Proof The proof is accomplished by analytic continuation of the results in Theorem 4.2, or in effect, by replacing ξ by ξ + iπ/2.
By exploiting the Q → P reduction (2.7), one easily obtains additional formulas, for

Tetrahedral Formulas (Schwarz Class III)
The following theorems give parametrized formulas for the Legendre/Ferrers func- The triple of exponent differences, (μ, μ, 2ν + 1), is respectively equal to , both lying in Schwarz's tetrahedral class III.

Theorem 5.1 The formulas
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 ∈ (−∞, ∞).

Proofs of Results in Sect. 3.1
The octahedral and tetrahedral formulas in Sects. 3.2, 4, and 5 followed from the theorems in Sect. 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 x = R(u) in Definition 3.1.
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 q v (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 = , which are the roots of q e := (1 + s 4 ) (1 − 34s 4 + s 8 ) = 1 − 33s 4 − 33s 8 + s 12 . Similarly, its eight face-centers f 1 , . . . f 8 , when radially projected, are located at s = (±1 ± i)(1 ± √ 3)/2, which are the roots of q f (s) := 1 + 14s 4 + s 8 . The polynomials q v , q e , 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 1 − q 3 f /q 2 e ; i.e., 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 f (x) is the Frobenius solution associated with 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 . . , 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 2 F 1 − 1 24 , 11 24 ; 3 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 equatioñ L m nf = 0, as one sees from (6.4), is far more complicated than D 2 sf = 0 (Laplace's equation) when (n, m) = (0, 0). 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 p v = 0. The roots u = 0, 1 of p v , 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 , −1, (3 − 2 √ 2) 2 of p e correspond to the four edgemidpoints 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 face-centers 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 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 with the zero characteristic exponent of the singular point u = 0. It satisfies eight differential recurrences of the form in which Δ(n, m) = (0, ±1), (±1, 0) and (±1, ±1). For each recurrence, the exponents σ v , σ e , σ f , the exponents ε v , ε e , ε f , and the prefactor K are listed in Table 1. The first four recurrences in the table, with Δ(n, m) = (0, ±1), (±1, 0), come a bit less easily, because they shift the 2 F 1 parameters in Definition 3.1 by half-integers rather than integers. But by examination, Definition 3.1 is equivalent to 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 Δ(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 nondifferential 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 Sect. 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 Δ(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 Sect. 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. Theorem 6.3 For any n ∈ Z, the octahedral function r 0 n has the Heun representation r 0 n (u) = Hn ; −3n, − 1 4 − 3n; 3 4 , −3n; −(2 + √ 3) 2 u and the equivalent expansion ∞ k=0 a k u k , where {a k } ∞ k=0 satisfy the second-order recurrence with a 0 = 1, a −1 = 0. Thus when n ≥ 0, r 0 n is a degree-3n Heun polynomial.
Proof If m = 0, the u = 1 singular point of L m n drops out, i.e., becomes ordinary, and .
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. Theorem 6.4 For any (n, m) ∈ Z 2 , the octahedral function r m n has the expansion r m n (u) = ∞ k=0 a k u k , where {a k } ∞ k=0 satisfy the third-order recurrence with a 0 = 1, a −1 = 0, a −2 = 0. Thus when n, m ≥ 0, r m n (u) is a degree-(3n + 2m) generalized Heun polynomial.
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 Δ(n, m) = (+1, 0), as in the proof of Theorem 6.2.
In the same way, the third and fourth formulas follow from the Δ(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 with the exponent 0 at u = 0, r m n is the Frobenius solution associated with 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

Biorthogonality of Octahedral Functions
The octahedral functions r m n = r m n (u), which are polynomials if n, m ≥ 0, satisfy recurrences, such as the three-term ones of Theorem 3.3, that are quite unlike the ones satisfied by the classical orthogonal polynomials. But at least if m = 0, −1, it can be shown that the family {r m n } m∈Z displays orthogonality on the u-interval [0, 1], or rather a form of biorthogonality.
The biorthogonality is best expressed in terms of the lifted functionsr m n (s) := r m n (u = s 4 ) of the last section, the full domain of which is the Riemann s-sphere in which the defining octahedron is inscribed. These are solutions ofL m nf = 0, 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 ν 0 +2n (z), n ∈ Z), and the eigenfunctions of the adjoint boundary value problem (i.e., P −μ ν 0 +2n (−z), n ∈ Z). The first problem is non-self-adjoint because the boundary conditions that single out P μ ν 0 +2n (z), n ∈ Z, as eigenfunctions are not selfadjoint.
However, one feature of the operatorL m n must be mentioned. Iff =f (s) solves L 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. 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 (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 Sects. 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 Sect. 8.1, and how dihedral Ferrers functions can be used for expansion purposes is explained in Sect. 8.2.

Explicit Formulas
The Jacobi polynomials P (α,β) n (z) are well known [11,Sect. 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.

Theorem 8.1 The formulas
hold when n is a nonnegative integer, for z ∈ (1, ∞) and ξ ∈ (0, ∞). (In the degenerate case when μ − n is a positive integer, P Proof Compare the representations (2.8a) and (8.1a).
(In the degenerate case when μ − n is a positive integer, P Proof The proof is accomplished by analytic continuation of Theorem 8.1, or in effect, by letting ξ = iθ . By exploiting theQ → P and Q → P reductions (2.6) and (2.7), one can derive additional formulas from Theorems 8.1 and 8.2, forQ respectively. However, the coefficients in (2.6) and (2.7) diverge when μ ∈ Z. Hence, following this approach to formulas forQ m , 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,Sect. 3.6.1]. Perhaps the best method is to expressQ m in terms of a 2 F 1 by using (2.8d), and then use known formulas for logarithmic 2 F 1 's [5].
In these formulas, the proportionality ofQ to each other is expected; cf. (2.4).
Also, the division in the 'minus' case by hold when m is a nonnegative 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.

Theorem 8.5 The formulas
hold when m is a nonnegative 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, we show that the result of [32] on the convergence of Love-Hunter expansions can be slightly extended.
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,Sect. 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 2 j+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) = u f (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 ν 0 +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 μ 0 +m ν 0 +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 finitevalued.) 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 skew-symmetric matrices can be represented by differential operators on S 2 , with real coefficients, and acts on the space of spherical harmonics. The resulting infinite-dimensional 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,Sect. 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 Sect. 9.1, the differential and nondifferential recurrences on ν and μ are derived. (See Theorems 9.1 and 9.2.) In Sect. 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 the setting of special function identities, which typically involve real linear combinations of differential operators, so(3, 2) arises more naturally than does so(4, 1).
In Sect. 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 half-odd-integer degree and order, have irreducible constituents that are identified as the known Dirac singleton representations of so(3, 2).

Differential and Nondifferential Recurrences
In any family {P μ 0 +m ν 0 +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.
Proof The four nondiagonal recurrences on the order and degree, with Δ(ν, μ) = ±(0, 1) and ±(1, 0), are classical and can be found in many reference works [11,31,36]. They can be deduced from the differential recurrences of Jacobi, which increment or decrement the parameters of the function 2 F 1 (a, b;  The final four diagonal ones, at least for P μ ν when ν, μ are integers, are due to Celeghini and del Olmo [4]. Each can be derived from the nondiagonal ones by a tedious process of elimination, but the process can be systematized as the calculation of the commutator of two differential operators. (See Sect. 9.2, below.) Table 2 Parameters for the differential recurrences of Theorem 9.1 In the rightmost column, the notation [a|b] ± signifies a, resp. b, in the +, resp. − case The differential recurrences satisfied by P μ ν can be written in circular-trigonometric forms that will be needed below. Substituting z = cos θ yields  ν is unaffected by the negating of the shifted degree parameter ν + 1 2 . The three-term ladder recurrences derived from the four pairs of differential recurrences are given in the following theorem. The diagonal ones, coming from the ladders with Δ(ν, μ) = ±(1, 1) and ±(1, −1), appear to be new.

Theorem 9.2
The Ferrers functions P μ ν = P μ ν (z) satisfy second-order (i.e., threeterm) recurrences on the order μ and degree ν, namely and the two diagonal recurrences The second-kind functions Q μ ν satisfy identical second-order recurrences. The Legendre functions P μ ν , Q μ ν (the latter unnormalized, as above), satisfy recurrences obtained from the preceding by (i) multiplying each term containing a function of order μ + δ and a coefficient proportional to [ √ 1 − z 2 ] α by a sign factor, equal to i α−δ , and (ii) replacing √ 1 − z 2 by √ z 2 − 1.
Proof Eliminate the derivative terms from the recurrences of Theorem 9.1. This is the procedure used to derive Gauss's three-term, nearest-neighbor 'contiguous function relations' for 2 F 1 (a, b; c; x) from Jacobi's differential recurrences on a, b; c.
It was noted in Sect. 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.
The last two turn out to commute. The real Lie algebra generated by J ± , K ± , of which these copies of so(3, R) and so(2, 1) are subalgebras, is 10-dimensional and is spanned over R by J ± , J 3 ; K ± , K 3 ; R ± ; S ± . It of course has real structure constants. For any (ν 0 , μ 0 ), its representation by differential operators on R 3 , as above, is carried by the real span of the solid harmonics S μ 0 +m ν 0 +n , (n, m) ∈ Z 2 . This result was obtained by Celeghini and del Olmo [4], though they confined themselves to integer ν, μ, i.e., in effect to (ν 0 , μ 0 ) = (0, 0). 1 To identify this 10-dimensional real algebra, it is useful to relabel its basis elements. First, let in each of which the three elements commute. The algebra can then be viewed as the span over R of J ± , J 3 ; P ± , P 3 ; C ± , C 3 and K 3 , which will be written as D henceforth. Define Also, for X = J, P, C, PC + , PC − , define the 'skew-Cartesian' elements X 1 := (X + + X − )/2, X 2 := (X + − X − )/2, so that X ± = X 1 ±X 2 . The algebra will then be the real span of J 1 , J 2 , J 3 ; P 1 , P 2 , P 3 ; C 1 , C 2 , C 3 ; D, or equivalently of J 1 , J 2 , J 3 ; PC ± 1 , PC ± 2 , PC ± 3 ; D. It is readily verified that J i commutes with PC + i and PC − i for i = 1, 2, 3, and that for i = 1, 2, 3. These identities specify the structure of the algebra. Now, recall that the real Lie algebra so( p, q) with p + q = n has the following defining representation. If Γ = (g i j ) = diag (+1, . . . , +1, −1, . . . , −1), with q +1's and p −1's, then so( p, q) comprises all real n × n matrices A for which Γ A is skewsymmetric. There is a sign convention here, and a p ↔ q symmetry; without loss of generality, p ≥ q will be assumed. It is sometimes useful to permute the +1's and −1's.
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 − g ac M bd − g bd M ac (9.11) are easily checked.
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'.
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).
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 x 2 1 + x 2 2 − x 2 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.

Lie Algebra Representations
In Sect. 9.2, it was shown that for any (ν 0 , μ 0 ), there are representations of the real Lie algebras so (3,2), so(4, 1), so(5, R) that are carried by the span of the family of (generically multi-valued) solid harmonics S μ ν (r, θ, ϕ), (ν, μ) ∈ (ν 0 , μ 0 ) + Z 2 . These arise from the action of the ladder operators on the Ferrers functions P where E α is the root vector associated with root α. The commutators [E α , 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]: