Hypergeometric type functions and their symmetries

We give a systematic and unified discussion of various classes of hypergeometric type equations: the hypergeometric equation, the confluent equation, the F_1 equation (equivalent to the Bessel equation), the Gegenbauer equation and the Hermite equation. In particular, we discuss recurrence relations of their solutions, their integral representations and discrete symmetries.

The theory of hypergeometric type functions is one of the oldest and most useful chapters of mathematics. In usual presentations, it appears complicated and messy. The main purpose of this paper is an attempt to present its basics in a way that shows clearly its internal structure and beauty.

Hypergeometric Type Functions
After the analysis of hypergeometric type operators, we discuss hypergeometric type functions, that is, functions annihilated by hypergeometric type operators. In particular, we will distinguish the so-called standard solutions which have a simple behavior around a singular point of the equation. In particular, if z 0 is a regular singular point, the Frobenius method gives us two solutions behaving as (z − z 0 ) λi , where λ 1 , λ 2 are the indices of z 0 . One can often find solutions with a simple behavior also around irregular singular points. For reflection invariant classes (5) and (6), one can also define another pair of natural solutions: the even solution S + , which we normalize by S + (0) = 1, and the odd solution S − , which we normalize by (S − ) (0) = 2.
Discrete symmetries can be used to derive properties of hypergeometric type functions. For instance, (1.6) implies that if f (z) solves the confluent equation for parameters c − a, c, then so does e z f (−z) for the parameters a, c. In particular, both functions F (a; c; z) and e z F (c−a; c; −z) solve the confluent equation for the parameters a, c. Both are analytic around z = 0 and equal 1 at z = 0. By the uniqueness of the solution to the Frobenius method, they should coincide. Hence, we obtain the identity F (a; c; z) = e z F (c − a; c; −z). (1.11) Commutation relations are also useful. For example, it follows immediately from (1.9) that (z∂ z + a)F (a; c; z) is a solution of the confluent equation for the parameters a + 1, c. At zero, it is analytic and its value is a. Hence we obtain the recurrence relation (z∂ z + a)F (a; c; z) = aF (a + 1; c; z). (1.12) For each class of equations, we describe a whole family of recurrence relations. Every such recurrence relation involves an operator of the following form: a 1st order differential operator with no dependence on the parameters + a multiplication operator depending linearly on the parameters. We will call them basic recurrence relations.
Sometimes, there also exist more complicated recurrence relations. We do not give their complete list, we only mention some of their examples. We call them additional recurrence relations.
Each of the standard solutions has simple integral representations of the form analogous to (1.4). Each of these integral representations is associated with a pair of (possibly infinite and possibly coinciding) points where the integrand has a singularity. We will use two basic kinds of contours for standard solutions: (a) The contour starts at one singularity and ends at the other singularity; we assume that at both singularities the analog of (1.5) is zero (hence, trivially, has equal values). (b) The contour starts at the first singularity, goes around the second singularity and returns to the first singularity; we assume that the analog of (1.5) is zero at the first singularity. If available, we will always treat the type (a) contour as the basic one.
For instance, under appropriate conditions on the parameters, the 1 F 1 function has the following two integral representations: type (a): F (a; c; z).
(0 + means that we bypass 0 in the counterclockwise direction; in this case, it is equivalent to bypassing ∞ in the clockwise direction).
There are various natural ways to normalize hypergeometric type functions. The most obvious normalization for a solution analytic at a given regular singular point is to demand that its value there is 1. (For the 2 F 0 equation, the point 0 is not regular singular; however, there is a natural generalization of this normalization condition). For equations (1)-(4), this function will be denoted by the letter F , consistently with the conventional usage. (Note the use of the italic font). In the case of reflection symmetric equations (5) and (6), we will use the letter S.
However, it is often preferable to use different normalizations, which involve appropriate values of the Gamma function or its products. Such normalizations arise naturally when we consider integral representations. They will be denoted by F for equations (1)-(4) (a similar notation can be found in [12]), and S for (5) and (6). (Note the use of the boldface roman font). Sometimes, there will be several varieties of these normalizations denoted by an appropriate superscript, related to various integral representations. The functions with these normalizations have often better properties than the F and S functions. This is especially visible in recurrence relations, where the coefficient on the right (such as a in (1.12)) depends on the normalization.
For example, for the 1 F 1 function, we introduce the following normalizations: F (a; c; z), the latter suggested by the type (a) integral representation given above.

Degenerate Case
For some values of parameters, hypergeometric type functions have special properties. This happens, in particular, when the difference of the indices at a given regular singular point is an integer. Then, the two standard solutions related to this point are proportional to one another. We call them degenerate solutions. (The best known example of such a situation is the Bessel functions of integer parameters). In this case, we have a simple generating function and an additional integral representation, which involves integrating over a closed loop.

Canonical Forms
Obviously, hypergeometric type operators coincide with differential operators of the form where σ is a polynomial of degree ≤ 2, κ is a polynomial of degree ≤ 1, and λ is a number. One can argue that it is natural to use σ, κ, λ to parametrize the hypergeometric type operators (more natural than σ, τ, η). Equation (1.13) will be denoted C(σ, κ, λ; z, ∂ z ), or, for brevity, C(σ, κ, λ). Let ρ(z) be a solution of the equation (1.14) (Note that Eq. (1.14) is solvable in elementary functions). We have the identity We will call ρ the natural weight. To justify this name note that if λ is real, σ, κ are real and ρ is positive and nonsingular on ]a, b[⊂ R, then C(σ, κ, λ) is Hermitian on the weighted space L 2 (]a, b[, ρ), when as the domain we take C ∞ c (]a, b[). It is sometimes useful to replace the operator C(σ, κ, λ) with We will call (1.16) the balanced form of C(σ, κ, λ). Sometimes, one replaces (1.1) by the 1-dimensional Schrödinger equation (1.17) is equivalent to (1.1), because Some of the symmetries of hypergeometric type equations are obvious in the balanced and Schrödinger-type forms. This is partly due to the fact that they do not change when we switch the sign in front of κ. This is a serious advantage of these forms.
In the literature, various forms of hypergeometric type equations are used. Instead of the Gegenbauer equation, one usually finds its balanced form, called the associated Legendre equation. The modified Bessel equation and the Bessel equation, equivalent to the rarely used 0 F 1 equation, are the balanced form of a special case of the 1 F 1 equation. Instead of the 1 F 1 equation, one often finds its Schrödinger-type form, the Whittaker equation. This usage, due mostly to historical traditions, makes the subject more complicated than necessary.
We will always use (1.1) as the basic form. Its main advantage is that in almost all cases the equation in the form (1.1) has at least one solution analytic around a given finite singular point. Even in the case of the 2 F 0 equation, whose all solutions have a branch point at 0, there exists a distinguished solution particularly well behaved at zero.

Hypergeometric Type Polynomials
Hypergeometric type polynomials, that is, polynomial solutions of hypergeometric type equations deserve a separate analysis. They have traditional names involving various nineteenth century mathematicians. Note, in particular, that the (rarely used) polynomial cases of the 2 F 0 function are called Bessel polynomials; however, they do not have a direct relation to the better-known Bessel functions.
There exists a well-known elegant approach to their theory that allows us to derive most of their basic properties in a unified way, see e.g. [10,13]. Let us sketch this approach.
Fix σ, κ, ρ, as in Sect. 1.5. For any n = 0, 1, 2, . . . , we define P n is a polynomial, typically of degree n, more precisely its degree is given as follows: 1. If σ = κ = 0, then deg P n = 0. We have a generating function an integral representation and recurrence relations In almost all sections, we devote a separate subsection to the corresponding class of polynomials. Beside the properties that follow immediately from the unified theory presented above, we describe additional properties valid in a given class.
The 0 F 1 equation does not have polynomial solutions, hence the corresponding section is the only one without a subsection about polynomials.
Another special situation arises in the case of the Gegenbauer equation. The standard Gegenbauer polynomials found in the literature do not have the normalization given by the Rodriguez-type formula. The Rodriguez-type formula yields the Jacobi polynomials, which for α = β coincide with the Gegenbauer polynomials up to a nontrivial coefficient. Thus, for the Gegenbauer equation, it is natural to consider two classes of polynomials differing by normalization. This is related to an interesting symmetry called the Whipple transformation, which is responsible for two kinds of integral representations.

Parametrization
Each class (1)-(6) depends on a number of complex parameters, denoted by Latin letters belonging to the set {a, b, c}. They will be called the classical parameters. They are convenient when we discuss power series expansions of standard solutions.
Unfortunately, the classical parameters are not convenient to describe discrete symmetries. Therefore, for each class (1)-(6), we introduce an alternative set of parameters, which we will call the Lie-algebraic parameters. They will be denoted by Greek letters such as α, β, μ, θ, λ, and will be given by certain linear (possibly, inhomogeneous) combinations of the classical parameters. Discrete symmetries of hypergeometric type equations will simply involve signed permutations of the Lie algebraic parameters-in the classical parameters, they look much more complicated. Recurrence relations also become simpler in the Lie-algebraic parameters.
For polynomials of hypergeometric type a third kind of parametrization is traditionally used. They are characterized by their degree n, which coincides with −a, where a is one of the classical parameters. The Lie-algebraic parameters appearing inside the 1st order part of the equation are used as the remaining parameters.
Let us stress that all these parametrizations are natural and useful. Therefore, we sometimes face the dilemma which parametrization to use for a given set of identities. We usually try to choose the one that gives the simplest formulas.
We sum up the information about various parametrizations in the following table:

Equation classical parameters
Lie-algebraic parameters Polynomial parameters for polynomials Hermite n = −a

Group-Theoretical Background
Identities for hypergeometric type operators and functions have a high degree of symmetry. Therefore, it is natural to expect that a certain group-theoretical structure is responsible for these identities. There exists a large literature about the relations between special functions and the group theory [6,8,15,16]. Nevertheless, as far as we know, the arguments found in the literature give a rather incomplete explanation of the properties that we describe. In a separate publication [2], we would like to present a group-theoretical approach to hypergeometric type functions with, we believe, a more satisfactory justification of their high symmetry. Below, we would like to briefly sketch the main ideas of [2].
Each hypergeometric type equation can be obtained by separating the variables of a certain 2nd order PDE of the complex variable with constant coefficients. One can introduce the Lie algebra of generalized symmetries of this PDE. In this Lie algebra, we fix a certain maximal commutative algebra, which we will call the "Cartan algebra". Operators whose adjoint action is diagonal in the "Cartan algebra" will be called "root operators". Automorphisms of the Lie algebra leaving invariant the "Cartan algebra" will be called "Weyl symmetries".
(Note that in some cases, the Lie algebra of symmetries is simple, and then the names Cartan algebra, root operators and Weyl symmetries correspond to the standard names. In other cases the Lie algebra is non-semisimple, and then the names are less standard-this is the reason for the quotation marks that we use). Now the parameters of hypergeometric type equation can be interpreted as the eigenvalues of elements of the "Cartan algebra". In particular, the Lie algebraic parameters correspond to a certain natural choice of the "Cartan algebra". Each recurrence relation is related to a "root operator". Finally, each symmetry of a hypergeometric type operator corresponds to a Weyl symmetry of the Lie algebra.
We can distinguish three kinds of PDE's with constant coefficients: 1. The Helmholtz equation on C n given by Δ n + 1, whose Lie algebra of symmetries is C n so(n, C); 2. The Laplace equation on C n given by Δ n , whose Lie algebra of generalized symmetries is so(n + 2, C) 3. The heat equation on C n ⊕ C given by Δ n + ∂ s , whose Lie algebra of generalized symmetries is sch(n, C) (the so-called (complex) Schrödinger Lie algebra.
Separating the variables in these equations usually leads to differential equations with many variables. Only in a few cases, it leads to ordinary differential equations, which turn out to be of hypergeometric type. Here is a table of these cases:

PDE
Lie algebra dimension of Cartan algebra discrete symmetries equation Hermite.

Comparison with the Literature
There exist many works that discuss hypergeometric type functions, e.g. [1,4,7,[11][12][13][14]17]. Some of them are meant to be encyclopedic collections of formulas, others try to show mathematical structure that underlies their properties. In our opinion, this work differs substantially from the existing literature. In our presentation, we try to follow the intrinsic logic of the subject, without too much regard for the traditions. If possible, we apply the same pattern to each class of hypergeometric type equations. This sometimes forces us to introduce unconventional notation.
We believe that the intricacy of usual presentations of hypergeometric type functions can be partly explained by historical reasons. In the literature, various classes of these functions are often described with help of different conventions. Sometimes, we will give short remarks devoted to the conventions found in the literature. These remarks will always be clearly separated from the main text.
Of course, our presentation does not contain all useful identities and properties of hypergeometric functions. Some of them are on purpose left out, e.g. the so-called addition formulas. We restrict ourselves to what we view as the most basic theory. On the other hand, we try to be complete for each type of properties that we consider.
Our work is strongly inspired by the book by Nikiforov and Uvarov [10], who tried to develop a unified approach to hypergeometric type functions. They stressed, in particular, the role of integral representations and of recurrence relations.
Another important influence is the works of Miller [8,9] who stressed the Lie-algebraic structure behind the recurrence relations.
The method of factorization can be traced back at least to [5].

Preliminaries
In this section, we fix basic terminology, notation and collect a number of well known useful facts, mostly from complex analysis. It is supposed to serve as a reference and can be skipped at the first reading.

Differential Equations
The main objectives of our paper are ordinary homogeneous 2nd order linear differential equations in the complex domain, that is equations of the form It will be convenient to treat (2.1) as the problem of finding the kernel of the operator We will then say that the Eq. (2.1) is given by the operator (2.2). When we do not consider the change of the variable, we will often write A for A(z, ∂ z ).

The Principal Branch of the Logarithm and the Power Function
The function is bijective. Its inverse will be called the principal branch of the logarithm and will be denoted simply log z.
If μ ∈ C then the principal branch of the power function is defined as Consequently, if α ∈ C\{0}, then the functions log(α(z − z 0 )) and Of course, if needed we will use the analytic continuation to extend the definition of the logarithm and the power function beyond C\] − ∞, 0] onto the appropriate covering of C\{0}.

Contours
We will write To avoid making pictures, we will use special notation for contours of integration.
Broken lines will be denoted as in the following example: This contour may be inappropriate if the function has a nonintegrable singularity at u. Then, we might want to bypass u with a small arc counterclockwise or clockwise. In such a case, we can use the curves We may want to bypass a group of points, say u 1 , u 2 . Such contours are denoted by A counterclockwise/clockwise loop around a group of points, say, A half-line starting at u and inclined at the angle φ is denoted We will also need slightly more complicated contours: Here, the contour departs from u at the angle φ, then it bypasses u with a small arc counterclockwise and then it goes in the direction of w.
The following contour has the shape of a kidney: Vol. 15 (2014)

Hypergeometric Type Functions and Their Symmetries 1585
This contour departs from u at the angle φ, then it goes around u and returns to u again at the angle φ. Instead of u + e i0 · 0, we will write u + 0. Likewise, instead of u + e iπ · 0, we will write u − 0.

Reflection Invariant Differential Equations
Consider a 2nd order differential operator Assume that (2.7) is invariant w.r.t. the reflection z → −z. This means that for some functions π, ρ, we have Then it is natural to make a quadratic change of coordinates: Thus, if g + (u), resp. g − (u) satisfy then g + (z 2 ) is an even solution, resp. zg − (z 2 ) is an odd solution of the equation given by (2.7). Note that if π, ρ are holomorphic, then 0 is a regular singular point of (2.8) with indices 0, 1 2 and of (2.9) with indices 0, − 1 2 .

Regular Singular Points
In this subsection, we recall well-known facts about regular singular points of differential equations We will write if f is analytic in a neighborhood of z 0 and f (z 0 ) = 1. Ann. Henri Poincaré An equation given by the operator with meromorphic coefficients a(z), c(z) has a regular singular point at The case λ 1 − λ 2 ∈ Z is called the degenerate case. In this case, the Frobenius method gives one solution corresponding to the point z 0 .
Likewise, (2.10) has a regular singular point at ∞ if Theorem 2.2 (The Frobenius method at infinity). If −λ 1 +λ 2 = −1, −2, . . ., then there exists a unique solutionf 1 (z) of (2.10) such thatf 1 Note the identity If z 0 is a regular singular point, then the corresponding indices of (2.11) equal those of (2.10) +θ. Likewise, if ∞ is a regular singular point, then the corresponding indices are shifted by −θ. The indices corresponding to other points are left unchanged.

The Gamma Function
In this section, we collect basic identities related to Euler's Gamma function that we will use.

The Pochhammer Symbol
If a ∈ C and n ∈ Z, then the so-called Pochhammer symbol is defined as follows: Note the identities 3. The 2 F 1 or the Hypergeometric Equation

Introduction
Let a, b, c ∈ C. Traditionally, the hypergeometric equation is given by the operator The classical parameters a, b, c will be often replaced by another set of parameters α, β, μ ∈ C, called Lie-algebraic. They are related to one another by In the Lie-algebraic parameters, the hypergeometric operator (3.1) becomes (3. 2) The Lie-algebraic parameters have an interesting interpretation in terms of the natural basis of the Cartan algebra of the Lie algebra so(6) [2]. The singular points of the hypergeometric operator are located at 0, 1, ∞. All of them are regular singular. The indices of these points are Thus, the Lie-algebraic parameters are the differences of the indices. The hypergeometric operator remains the same if we interchange a and b (replace μ with −μ).

Integral Representations
Analogous (and nonequivalent) integral representations can be obtained by interchanging a and b in Theorem 3.1.

Symmetries
To every permutation of the set of singularities {0, 1, ∞}, we can associate exactly one homography z → w(z). Using the method described at the end of Sect. 2.5, with every such homography we can associate 4 substitutions that preserve the form of the hypergeometric equation. Altogether there are 6 × 4 = 24 substitutions. They form a group isomorphic to the group of proper symmetries of the cube. If we take into account the fact that replacing μ with −μ is also an obvious symmetry of the hypergeometric equation, then we obtain a group of 2 × 24 = 48 elements, isomorphic to the group of all (proper and improper) symmetries of a cube, which is the Weyl group of so (6).
Below, we describe the table of symmetries of the hypergeometric operator except for those obtained by switching the sign of the last parameter. We fix the sign of the last parameter by demanding that the number of minus signs is even.
Note that the table looks much simpler in the Lie-algebraic parameters than in the classical parameters. All the operators below equal F α,β,μ (w, ∂ w ) for the corresponding w:

Factorization and Commutation Relations
The hypergeometric operator can be factorized in several ways: One way of showing the above factorizations is as follows: We start with deriving the first one, and then we apply the symmetries of Sect. 3.3. The factorizations can be used to derive the following commutation relations: Each of these commutation relations corresponds to a root of the Lie algebra so(6).

Canonical Forms
The natural weight of the hypergeometric operator is z α (1 − z) β , so that Vol. 15 (2014) Hypergeometric Type Functions and Their Symmetries 1593 The balanced form of the hypergeometric operator is Note that the symmetries α → −α, β → −β and μ → −μ are obvious in the balanced form.

The Hypergeometric Function
defined for all a, b, c ∈ C. Another useful function proportional to 2 F 1 is It has the integral representation It is proven essentially in the same way as (3.5), except that instead of (2.19) we use (2.21). We have the identities In fact, by the 3rd, 9th and 11th symmetry of Sect. 3.3, all these functions are annihilated by the hypergeometric operator. All of them are ∼ 1 at 1. Hence, by the uniqueness of the Frobenius method they coincide, at least for c = 0, −1, . . . . By continuity, the identities hold for all c ∈ C. Let us introduce new notation for various varieties of the hypergeometric function involving the Lie-algebraic parameters instead of the classical parameters.

Standard Solutions: Kummer's Table
To each of the singular points 0, 1, ∞, we can associate two solutions corresponding to its indices. Thus, we obtain 3 × 2 = 6 solutions, which we will call standard solutions. Using the identities (3.7), each solution can be written in 4 distinct ways (not counting the trivial change of the sign in front of the last parameter). Thus, we obtain a list of 6 × 4 = 24 expressions for solutions of the hypergeometric equation, called sometimes Kummer's table.
We describe the standard solutions to the hypergeometric equation in this section. We will use consistently the Lie-algebraic parameters, which give much simpler expressions.
It follows from Theorem 3.1 that for appropriate contours γ integrals of the form An integral representation for Re(1 + α) > |Re(β − μ)|: Note that all the identities of this subsubsection are the transcriptions of identities of Sect. 3.6 to the Lie-algebraic parameters.

Solution
∼ z −α at 0. If α = 1, 2, . . . , then the following function is the unique solution behaving as z −α at 0: Integral representations for Re(1 − α) > |Re(β − μ)|: To check these identities, we note first that the integrals are solutions of the hypergeometric equation. By substituting t = zs, we easily check that they have the correct behavior at zero. Of course, it is elementary to pass from the first identity, which is adapted to the region on the right of the singularity z = 0 to the second, adapted to the region on the left of the singularity. For convenience, we give both identities.

Connection Formulas
We use the solutions ∼ 1 and ∼ z −α at 0 as the basis. We show how the other solutions decompose in this basis. For the first pair of relations, we assume that z ∈] − ∞, 0]∪[1, ∞[:

J. Dereziński Ann. Henri Poincaré
For the second pair, we assume that z ∈ [0, ∞[ The connection formulas are easily derived from the integral representations by looking at the behavior around 0.

Recurrence Relations
The following recurrence relations follow easily from the commutation relations of Sect. 3.4:

Additional Recurrence Relations
There exist other, more complicated recurrence relations for hypergeometric functions, for example Note that (3.9) follows from the 6th and 7th recurrence relation, and (3.10) follows from the 5th and 8th of Sect. 3.9.

Degenerate
Case α = m ∈ Z is the degenerate case of the hypergeometric equation at 0. We have then This easily implies the identity Thus, the two standard solutions determined by the behavior at zero are proportional to one another. One can also see the degenerate case in the integral representation (3.3). If we go around 0, z, the phase of the integrand changes by e i2πc = e i2πα . Therefore, if α = m ∈ Z, then the loop around 0, z is closed on the Riemann surface of the integrand. We have an additional integral representation and a generating function: 1 2πi To see the integral representation, we note that the integral on the l.h.s. is annihilated by the hypergeometric operator. Then, we check that its value at zero equals 1 2πi The second identity follows from (3.11). Another way to see it is to make the substitution t = z s . Note that [(0, z) + ] becomes [(∞, 1) + ], which coincides with [(0, z) − ]. Then, we change the sign in front of the integral and the orientation of the contour of integration, obtaining Finally, we apply the first integral representation again.
The generating function follows from the integral representation.

Jacobi Polynomials
If −a = n = 0, 1, . . . , then hypergeometric functions are polynomials. We will call them the Jacobi polynomials. Following Sect. 1.6, the Jacobi polynomials are defined by the Rodriguez-type formula Remark 3.3. In most of the literature, the Jacobi polynomials are slightly different: The equation: 0 = F(−n, 1 + α + β + n; β + 1; z, ∂ z )P α,β n (z) Vol. 15 (2014) Hypergeometric Type Functions and Their Symmetries 1601 Generating functions: Integral representations: Discrete symmetries: Recurrence relations: The first, second, resp. third integral representation is easily seen to be equivalent to the first, second, resp. third generating function. The first follows immediately from the Rodriguez-type formula.
The symmetries can be interpreted as a subset of Kummer's table. The first line corresponds to the symmetries of the solution regular at 0, see (3.7) (or Sect. 3.7.1). Note that from 4 expressions in (3.7) only the first and the third survive, since n = −a should not change. The second line corresponds to the solution regular at 1 (Sect. 3.7.3), finally the third line to the solution ∼ z −a = z n (Sect. 3.7.5).
The differential equation, the Rodriguez-type formula, the first generating function, the first integral representation and the first pair of recurrence relations are special cases of the corresponding formulas of Sect. 1.6.
Note that Jacobi polynomials are regular at 0, 1, and behave as z n in infinity. Thus (up to coefficients), they coincide with the 3 standard solutions. They have the following values at 0, 1 and the behavior at ∞: We have several alternative expressions for Jacobi polynomials: One way to derive the first of the above identities is to use integral representation (3.6). Using that a is an integer we can replace the open curve Then, making the substitutions s = z − 1 t , s = zt, resp. s = (z − 1)t, we obtain the 1st, 2nd, resp. 3rd integral representation.
Additional identities valid in the degenerate case: There is a region where Jacobi polynomials are zero. This happens iff α, β ∈ Z and α, β are in the triangle In the analysis of symmetries of Jacobi polynomials, it is useful to go back to the Lie-algebraic parameters, more precisely, to set μ := −α − β − 2n − 1. Then (3.12) acquires a more symmetric form, since we can replace its last condition by 0 ≤ μ + n.
One can distinguish three strips where Jacobi polynomials have special properties. Note that the intersection of the strips below is precisely the triangle described in (3.12).

Special Cases
Beside the polynomial and degenerate cases, the hypergeometric equation has a number of other special cases. In their description most of the time, we will use the Lie-algebraic parameters, which are here more convenient than the classical parameters.

Gegenbauer Equation Through an Affine Transformation.
Consider a hypergeometric equation whose two parameters coincide up to a sign. After applying an appropriate symmetry, we can assume that they are at the first and second place, and that they are equal to one another. In other words, α = β. A simple affine transformation (6.2) can be then applied to obtain a reflection invariant equation called the Gegenbauer equation. We study it separately in Sect. 6.

Gegenbauer Equation Through a Quadratic Transformation.
Hypergeometric equations with one of the parameters equal to 1 2 or − 1 2 also enjoy special properties. After applying, if needed, one of the symmetries, we can assume that μ = ± 1 2 . Then identity (6.4) or (6.5) leads to the Gegenbauer equation.

Chebyshev Equation.
Even more special properties have equations with a pair of parameters ± 1 2 . After applying one of the symmetries, we can assume that α = β = 1 2 . Thus, we are reduced to the Chebyshev equation of the first kind; see (6.15). Another option is to reduce it to the Chebyshev equation of the second kind, which corresponds to α = β = − 1 2 ; see (6.16).

Legendre Equation.
Let L be the sublattice of Z 3 consisting of points whose sum of coordinates is even. It is a sublattice of Z 3 of degree 2. Using recurrence relations of Sect. 3.9, we can pass from hypergeometric functions with given Lie-algebraic parameters (α, β, μ) to parameters from (α, β, μ) + L. This is especially useful in the degenerate case, when some of the parameters are integers. In particular, if two of the parameters are integers, by applying recurrence relations we can make both of them zero. By applying an appropriate symmetry, we can assume that α = β = 0. Thus, we obtain the Legendre equation, see (6.14).

Fully Degenerate
Case. An interesting situation arises if α, β, μ ∈ Z, that is, we have the degenerate case at all singular points. We can distinguish two situations: 1. If α + β + μ is even, by walking on the lattice L we can reduce ourselves to the equation for the complete elliptic integral, which corresponds to α = β = μ = 0. 2. If α + β + μ is odd, by walking on the lattice L we can reduce ourselves to the equation for the Legendre polynomial of degree 0, which corresponds to α = β = 0, μ = 1. This equation is solved by F 0,0,1 (z) = F (0, 1; 1; z) = 1, where we used Kummer's table and This equation is a limiting case of the hypergeometric equation:

The 2 F 0 Equation
Parallel to the 1 F 1 equation, we will consider the 2 F 0 equation, given by the operator where a, b ∈ C. This equation is another limiting case of the hypergeometric equation:

Equivalence of the 1 F 1 and 2 F 0 Equation
Note that Hence the 2 F 0 equation is equivalent to the 1 F 1 equation. We will treat the 1 F 1 equation as the principal one. The relationship between the parameters is

Lie-Algebraic Parameters
Instead of the classical parameters, we usually prefer the Lie-algebraic parameters α, θ: In these parameters, the 1 F 1 operator (4.1) becomes and the 2 F 0 operator (4.2) becomes The Lie-algebraic parameters have an interesting interpretation in terms of a natural basis of a "Cartan algebra" of the Lie algebra sch(2) [2]. Vol. 15 (2014) Hypergeometric Type Functions and Their Symmetries 1607

Integral Representations
Two kinds of integral representations of solutions to the 1 F 1 equation are described below: Proof. We check that for any contour γ the l.h.s of (4.4) and (4.
For solutions of the 2 F 0 equation, we also have two kinds of integral representations: Proof. We check that for any contour γ (4.6) equals The second integral representation is obtained if we interchange a and b. 1608 J. Dereziński Ann. Henri Poincaré

Symmetries
The following operators equal F θ,α (w, ∂ w ) for the appropriate w: The third symmetry is sometimes called the 1st Kummer transformation.
Symmetries of the 1 F 1 operators can be interpreted as the "Weyl group" of the Lie algebra sch(2).

Factorizations and Commutation Relations
There are several ways of factorizing the 1 F 1 operator.
One can use the factorizations to derive the following commutation relations: Vol. 15 (2014) Hypergeometric Type Functions and Their Symmetries 1609 Each of these commutation relations can be associated with a "root" of the Lie algebra sch(2).

Canonical Forms
The natural weight of the 1 F 1 operator equals z α e −z , so that The balanced form of the 1 F 1 operator is which are the operators for the modified Bessel and Bessel equations. Thus, both these equations essentially coincide with the balanced form of the 1 F 1 equation with θ = 0. We will discuss them further in Remark 5.3.
The Schrödinger form of the 1 F 1 equation is Remark 4.4. In the literature, the equation given by (4.7) is often called the Whittaker equation. Its standard form is Thus, κ, μ correspond to − θ 2 , α 2 .
The natural weight of the 2 F 0 operator equals z θ e 1 z , so that The balanced form of the 2 F 0 operator is In the Lie-algebraic parameters:  where | arg c − π| < π − , > 0. It extends to an analytic function on the universal cover of C\{0} with a branch point of an infinite order at 0. It has the following asymptotic expansion: Sometimes instead of 2 F 0 , it is useful to consider the function We have an integral representation for Rea > 0 and without a restriction on parameters 1 2πi When we use the Lie-algebraic parameters, we denote the 2 F 0 function byF andF. The tilde is needed to avoid the confusion with the 1 F 1 function: It is one of solutions of the 1 F 1 equation, which we will discuss in Sect. 4.11.3. One also uses the Whittaker function of the 2nd kind which solves the Whittaker equation.

Standard Solutions
The 1 F 1 equation has two singular points. 0 is a regular singular point and with each of its two indices, we can associate the corresponding solution. ∞ is not a regular singular point. However, we can define two solutions with a simple behavior around ∞. Altogether we obtain 4 standard solutions, which we will describe in this subsection. It follows by Theorem 4.1 that, for appropriate contours γ 1 , γ 2 , the integrals In the first integral, the natural candidates for the endpoints of the intervals of integration are {−∞, 0, z}. We will see that all 4 standard solutions can be obtained as such integrals.
In the second integral the natural candidates for endpoints are {1, 0 − 0, ∞}. (Recall from Sect. 2.3 that 0 − 0 denotes 0 approached from the left). The 4 standard solutions can be obtained also from the integrals with these endpoints.

Solution
The first integral representation is valid for all parameters: The second is valid for Re(1 + α) > |Reθ|: Integral representation for Re(1 − α) > |Reθ|: and without a restriction on parameters: 1 2πi

Connection Formulas
We decompose standard solutions in pair of solutions with a simple behavior around zero.

Recurrence Relations
The following recurrence relations follow easily from the commutation relations of Sect. 4.7: The recurrence relations for the 2 F 0 functions are similar:

Additional Recurrence Relations
There exists an additional pair of recurrence relations: Thus, the two standard solutions determined by the behavior at zero are proportional to one another. One can also see the degenerate case in the integral representations: 1 2πi The corresponding generating functions are
Following Sect. 1.6, they can be defined by the following version of the Rodriguez-type formula: Ann. Henri Poincaré The differential equation: Generating functions: Integral representations: Expression in terms of the Bessel polynomials (to be defined in the next subsection):
The differential equation, the Rodriguez-type formula, the first generating function, the first integral representation and the first pair of recurrence relations are special cases of the corresponding formulas of Sect. 1.6.
Then, we set s = − 1 t , resp. s = 1 − 1 t to obtain the integral representations. The value at 0 and behavior at ∞: An additional identity valid in the degenerate case:

Bessel Polynomials
The 2 F 0 functions for −a = n = 0, 1, 2, . . . are polynomials. Appropriately normalized they are called Bessel polynomials. They are seldom used in the literature, because they do not form an orthonormal basis in any weighted space and they are easily expressed in terms of Laguerre polynomials. Following Sect. 1.6, they can be defined by the following version of the Rodriguez-type formula: Differential equation: Generating functions: Integral representations:

Symmetries
The only nontrivial symmetry is It can be interpreted as a "Weyl symmetry" of aso(2).

Factorizations and Commutation Relations
There are two ways to factorize the 0 F 1 operator: The factorizations can be used to derive the following commutation relations: Each commutation relation can be associated with a "root" of the Lie algebra aso(2).

Canonical Forms
The natural weight of the 0 F 1 operator is z α , so that The balanced form of the 0 F 1 operator is The symmetry α → −α is obvious in the balanced form.
Remark 5.3. In the literature, the 0 F 1 equation is seldom used. Much more frequent is the modified Bessel equation, which is equivalent to the 0 F 1 equation: where z = w 2 4 , w = ±2 √ z. Even more frequent is the Bessel equation: We can express the 0 F 1 function in terms of the confluent function It is also a limit of the confluent function. For Rec > 1 2 , we have a representation called the Poisson formula: We will usually prefer to use the Lie-algebraic parameters: F α (z) := F (α + 1; z), F α (z) := F(α + 1; z).
Remark 5.4. In the literature, the 0 F 1 function is seldom used. Instead, one uses the modified Bessel function and, even more frequently, the Bessel function:

Connection Formulas
We can use the solutions with a simple behavior at zero as the basis: Alternatively, we can use theF function and its analytic continuation around 0 in the clockwise or anti-clockwise direction as the basis:

Recurrence Relations
The following recurrence relations easily follow from the commutation relations of Sect. 5.5: (z∂ z + α) F α (z) = F α−1 (z). Thus, the two standard solutions determined by the behavior at zero are proportional to one another. We have an integral representation, called the Bessel formula, and a generating function: 1 2πi The natural endpoints of γ 2 are z + √ z 2 − 1, z − √ z 2 − 1, 0, ∞. Similarly, all standard solutions can be obtained from the integrals over contours with these endpoints.

Degenerate Case
It is interesting to note that in some aspects the theory of the Gegenbauer equation is more complicated than that of the hypergeometric equation. One of its manifestations is a relatively big number of natural normalizations of solutions. Indeed, let us consider, e.g., integral representations of the type (6.12). The natural endpoints fall into two categories: {1, −1} and {0, ∞}. Therefore, we have three kinds of contours joining two of these endpoint: [−1, 1], [0, ∞[ and the contours joining two distinct categories. This corresponds two three distinct natural normalizations, which we describe in what follows.
We will also introduce several alternatively normalized functions: The initial conditions at 0 and the identities for the even and odd case are given only for C II,α n , since those for C I,α n are more complicated: We have the following special cases: 1. If α ∈ Z, −n ≤ α ≤ −n−1 2 , then C I,α n = 0.

Special Cases
When describing special cases of the Gegenbauer equation, we will primarily use the Lie-algebraic parameters.

The Legendre Equation.
Suppose that one of the parameters is an integer. Using, if necessary, recurrence relations, we can assume that it is zero. After applying an appropriate symmetry, we can assume that α = 0. We obtain then the Legendre operator: (6.14) For the particular case λ = 0, its solutions can be expressed by the so-called complete elliptic functions. The Legendre operator for polynomials of degree n has the form (1 − z 2 )∂ 2 z − 2z∂ z + n(n + 1).