# Spherical functions

solid spherical harmonics, associated Legendre functions of the first and second kinds

Two linearly independent solutions $P _ \nu ^ \mu ( z)$ and $Q _ \nu ^ \mu ( z)$ of the differential equation

$$( 1- z) ^ {2} \frac{d ^ {2} y }{dz ^ {2} } - 2z \frac{dy}{dz} + \left [ \nu ( \nu + 1) - \frac{\mu ^ {2} }{1- z ^ {2} } \right ] = 0,$$

where $\mu$ and $\nu$ are complex constants, which arises in the solution of a class of partial differential equations by the method of separation of variables (cf. Separation of variables, method of). The points $z = \pm 1 , \infty$ are branching points of the solutions, in general. The spherical functions are particular cases of the hypergeometric functions (cf. Hypergeometric function):

$$P _ \nu ^ \mu ( z) = \frac{1}{\Gamma ( 1- \mu ) } \left ( z+ \frac{1}{z-} 1 \right ) ^ {\mu /2 } {} _ {2} F _ {1} \left ( - \nu , \nu + 1; 1- \mu ; 1- \frac{z}{2} \right )$$

$$\left ( \mathop{\rm arg} z+ \frac{1}{z-} 1 = 0 \ \textrm{ if } \mathop{\rm Im} z = 0, z > 1 \right ) ,$$

$$Q _ \nu ^ \mu ( z) = \frac{e ^ {\mu \pi i } \sqrt \pi \Gamma ( \mu + \nu + 1) }{2 ^ {\nu + 1 } \Gamma ( \nu + 3/2) } \frac{( z ^ {2} - 1) ^ {\mu /2 } }{z ^ {\mu + \nu + 1 } } \times$$

$$\times {} _ {2} F _ {1} \left ( \frac{\mu + \nu + 1 }{2} , \frac{\mu + \nu + 2 }{2} ; \nu + \frac{3}{2} ; \frac{1}{z ^ {2} } \right )$$

$$\textrm{ ( } \mathop{\rm arg} z = 0 \ \textrm{ if } \mathop{\rm Im} z = 0, z > 0;$$

$$\mathop{\rm arg} ( z ^ {2} - 1) = 0 \ \textrm{ if } \mathop{\rm Im} z = 0, z > 1 \textrm{ ) }.$$

The spherical functions $P _ \nu ^ \mu ( z)$ and $Q _ \nu ^ \mu ( z)$ are defined and single-valued in the domains $| 1- z |< 2$ and $| z | > 1$, respectively, of the complex plane cut by the real axis from $- \infty$ to $+ 1$.

If $\mathop{\rm Im} z = 0$, $z = x$, $- 1 < x < 1$, then the following functions are usually taken as solutions:

$$P _ \nu ^ \mu ( z) = \frac{1}{2} [ e ^ {\mu \pi i/2 } P _ \nu ^ \mu ( x+ i0) + e ^ {- \mu \pi i/2 } P _ \nu ^ \mu ( x- i0) ] =$$

$$= \ \frac{1}{\Gamma ( 1- \mu ) } \left ( 1+ \frac{x}{1-} x \right ) ^ {\mu /2 } {} _ {2} F _ {1} \left ( - \nu , \nu + 1 ; 1- \mu ; 1- \frac{x}{2} \right ) ,$$

$$Q _ \nu ^ \mu ( z) = \frac{1}{2} e ^ {\mu \pi i } [ e ^ {- \mu \pi i/2 } Q _ \nu ^ \mu ( x+ i0) + e ^ {\mu \pi i/2 } Q _ \nu ^ \mu ( x- i0) ] =$$

$$= \ \frac \pi {2 \sin \mu \pi } \left [ \cos \mu \pi P _ \nu ^ \mu ( x) - \frac{\Gamma ( \nu + \mu + 1) }{\Gamma ( \nu - \mu + 1) } P _ \nu ^ {- \mu } ( x) \right ] ,$$

where $f( x+ i0)$ $( f( x- i0))$ are the values of the function $f( z)$ on the upper (lower) boundary of the cut.

When $\mu = 0$, $\nu = n = 0, 1 \dots$ $P _ {n} ( z) \equiv P _ {n} ^ {0} ( z)$ are the Legendre polynomials. For zonal spherical functions see Spherical harmonics.

#### References

 [1] H. Bateman (ed.) A. Erdélyi (ed.) et al. (ed.) , Higher transcendental functions , 2. Bessel functions, parabolic cylinder functions, orthogonal polynomials , McGraw-Hill (1953) [2] E. Jahnke, F. Emde, "Tables of functions with formulae and curves" , Dover, reprint (1945) (Translated from German) [3] E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952) pp. Chapt. 6 [4] A. Krazer, W. Franz, "Transzendente Funktionen" , Akademie Verlag (1960) [5] E.W. Hobson, "The theory of spherical and ellipsoidal harmonics" , Chelsea, reprint (1955)

A more common usage of the phrase "spherical function" is as follows.

Let $G$ be a unimodular locally compact group and $K$ a subgroup of $G$. Let $\pi$ be an irreducible unitary representation of $G$ on a Hilbert space ${\mathcal H}$ such that the $K$- fixed vectors in ${\mathcal H}$ form a one-dimensional subspace, spanned by a unit vector $e$. Then the $K$- bi-invariant function $\phi$ on $G$ defined by $\phi ( x) = ( e, \pi ( x) e)$ is called a spherical function. Sometimes $\phi$ is called a zonal spherical function, while the functions $x \rightarrow ( v, \pi ( x) e)$( $v \in {\mathcal H}$) are also called spherical functions. Some authors call $\phi$ an elementary spherical function, while all $K$- bi-invariant functions on $G$ are called spherical functions.

The pair $( G, K)$ is a Gel'fand pair if, for all irreducible unitary representations of $G$, the subspace of $K$- fixed vectors in the representation space has dimension $1$ or $0$. This is equivalent to the commutativity of the convolution algebra $C _ {c} ( K \setminus G/K)$ of $K$- bi-invariant continuous functions on $G$ with compact support. Now spherical functions are more generally defined as solutions $\phi$, not identically zero, of the functional equation

$$\tag{* } \phi ( x) \phi ( y) = \int\limits _ { K } \phi ( xky) dk,\ x, y \in G,$$

where $dk$ is the normalized Haar measure on $K$. These solutions include the spherical functions associated with irreducible unitary representations. Other solutions may be associated with irreducible non-unitary representations of $G$. The characters of the comutative algebra $C _ {c} ( K \setminus G/K)$ are precisely the mappings $f \rightarrow \int _ {G} f( x) \phi ( x) dx$, where $dx$ is Haar measure on $G$ and $\phi$ is a solution of (*).

If $G$ is, moreover, a connected Lie group, then $( G, K)$ is a Gel'fand pair if and only if the algebra ${\mathcal D} ( G/K)$ of $G$- invariant differential operators on the homogeneous space $G/K$ is commutative. Then $\phi$ is a solution of (*) if and only if it is $K$- bi-invariant, $C ^ \infty$, $\phi ( e)= 1$, and the function $xK \rightarrow \phi ( x)$ on $G/K$ is a joint eigenfunction of the elements of ${\mathcal D} ( G/K)$. In particular, if $G$ is a connected real semi-simple LIe group and $K$ is a maximal compact subgroup, then $( G, K)$ is a Gel'fand pair, $G/K$ is a Riemannian symmetric space, and much information is available about ${\mathcal D} ( G/K)$ and the sperical functions.

#### References

 [a1] J. Fauraut, "Analyse harmonique sur les paires de Gelfand et les espaces hyperboliques" , Anal. Harmonique , CIMPA (1982) pp. 315–446 [a2] I.M. Gel'fand, "Spherical functions on symmetric spaces" Transl. Amer. Math. Soc. , 37 (1964) pp. 39–44 Dokl. Akad. Nauk SSSR , 70 (1950) pp. 5–8 [a3] R. Godement, "Introduction aux traveaux de A. Selberg" Sem. Bourbaki , 144 (1957) [a4] S. Helgason, "Groups and geometric analysis" , Acad. Press (1984) pp. Chapt. II, Sect. 4
How to Cite This Entry:
Spherical functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spherical_functions&oldid=48775
This article was adapted from an original article by Yu.A. BrychkovA.P. Prudnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article