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)

Comments

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