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''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|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|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|Legendre polynomials]]. For zonal spherical functions see [[Spherical harmonics|Spherical harmonics]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Bateman (ed.)  A. Erdélyi (ed.)  et al. (ed.) , ''Higher transcendental functions'' , '''2. Bessel functions, parabolic cylinder functions, orthogonal polynomials''' , McGraw-Hill  (1953)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Jahnke,   F. Emde,   "Tables of functions with formulae and curves" , Dover, reprint  (1945)  (Translated from German)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.T. Whittaker,   G.N. Watson,   "A course of modern analysis" , Cambridge Univ. Press  (1952)  pp. Chapt. 6</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A. Krazer,   W. Franz,   "Transzendente Funktionen" , Akademie Verlag  (1960)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E.W. Hobson,   "The theory of spherical and ellipsoidal harmonics" , Chelsea, reprint  (1955)</TD></TR></table>

====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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Fauraut,   "Analyse harmonique sur les paires de Gelfand et les espaces hyperboliques" , ''Anal. Harmonique'' , CIMPA  (1982)  pp. 315–446</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Godement,   "Introduction aux traveaux de A. Selberg"  ''Sem. Bourbaki'' , '''144'''  (1957)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  S. Helgason,   "Groups and geometric analysis" , Acad. Press  (1984)  pp. Chapt. II, Sect. 4</TD></TR></table>