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''solid spherical harmonics, associated Legendre functions of the first and second kinds''
 
''solid spherical harmonics, associated Legendre functions of the first and second kinds''
  
Two linearly independent solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s0866701.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s0866702.png" /> of the differential equation
+
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]]):
 +
 
 +
$$
 +
P _  \nu  ^  \mu  ( z)  = 
 +
\frac{1}{\Gamma ( 1- \mu ) }
 +
\left ( z+
 +
\frac{1}{z-}
 +
1 \right ) ^ {\mu /2 } {} _ {2} F _ {1} \left ( - \nu , \nu + 1;  1- \mu ;  1-
 +
\frac{z}{2}
 +
\right )
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s0866703.png" /></td> </tr></table>
+
$$
 +
\left (  \mathop{\rm arg}  z+
 +
\frac{1}{z-}
 +
= 0 \ 
 +
\textrm{ if }  \mathop{\rm Im}  z = 0, z > 1 \right ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s0866704.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s0866705.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s0866706.png" /> are branching points of the solutions, in general. The spherical functions are particular cases of the hypergeometric functions (cf. [[Hypergeometric function|Hypergeometric function]]):
+
$$
 +
Q _  \nu  ^  \mu  ( z)  =
 +
\frac{e ^ {\mu \pi i } \sqrt \pi
 +
\Gamma ( \mu + \nu + 1) }{2 ^ {\nu + 1 } \Gamma ( \nu + 3/2) }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s0866707.png" /></td> </tr></table>
+
\frac{( z  ^ {2} - 1) ^ {\mu /2 } }{z ^ {\mu + \nu + 1 } }
 +
\times
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s0866708.png" /></td> </tr></table>
+
$$
 +
\times
 +
{} _ {2} F _ {1} \left (
 +
\frac{\mu + \nu + 1 }{2}
 +
,
 +
\frac{\mu +
 +
\nu + 2 }{2}
 +
; \nu +
 +
\frac{3}{2}
 +
;
 +
\frac{1}{z  ^ {2} }
 +
\right )
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s0866709.png" /></td> </tr></table>
+
$$
 +
\textrm{ (  }  \mathop{\rm arg}  z  = 0 \ 
 +
\textrm{ if }  \mathop{\rm Im}  z = 0, z > 0;
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667010.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm arg} ( z  ^ {2} - 1)  = 0 \ 
 +
\textrm{ if }  \mathop{\rm Im}  z = 0, z > 1 \textrm{ )  }.
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667011.png" /></td> </tr></table>
+
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 $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667012.png" /></td> </tr></table>
+
If  $  \mathop{\rm Im}  z = 0 $,
 +
$  z = x $,
 +
$  - 1 < x < 1 $,
 +
then the following functions are usually taken as solutions:
  
The spherical functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667014.png" /> are defined and single-valued in the domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667016.png" />, respectively, of the complex plane cut by the real axis from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667017.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667018.png" />.
+
$$
 +
P _  \nu  ^  \mu  ( z)  =
 +
\frac{1}{2}
 +
[ e ^ {\mu \pi i/2 } P _  \nu  ^  \mu
 +
( x+ i0) + e ^ {- \mu \pi i/2 } P _  \nu  ^  \mu  ( x- i0) ] =
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667021.png" />, then the following functions are usually taken as solutions:
+
$$
 +
= \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667022.png" /></td> </tr></table>
+
\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 ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667023.png" /></td> </tr></table>
+
$$
 +
Q _  \nu  ^  \mu  ( z)  =
 +
\frac{1}{2}
 +
e ^ {\mu \pi i } [ e ^ {- \mu \pi
 +
i/2 } Q _  \nu  ^  \mu  ( x+ i0) + e ^ {\mu \pi i/2 } Q _  \nu  ^  \mu  ( x- i0) ] =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667024.png" /></td> </tr></table>
+
$$
 +
= \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667025.png" /></td> </tr></table>
+
\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667026.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667027.png" /> are the values of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667028.png" /> on the upper (lower) boundary of the cut.
+
where $  f( x+ i0) $
 +
$  ( f( x- i0)) $
 +
are the values of the function $  f( z) $
 +
on the upper (lower) boundary of the cut.
  
When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667030.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667031.png" /> are the [[Legendre polynomials|Legendre polynomials]]. For zonal spherical functions see [[Spherical harmonics|Spherical harmonics]].
+
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====
 
====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>
 
<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====
 
====Comments====
 
A more common usage of the phrase  "spherical function"  is as follows.
 
A more common usage of the phrase  "spherical function"  is as follows.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667032.png" /> be a unimodular locally compact group and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667033.png" /> a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667034.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667035.png" /> be an irreducible unitary representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667036.png" /> on a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667037.png" /> such that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667038.png" />-fixed vectors in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667039.png" /> form a one-dimensional subspace, spanned by a unit vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667040.png" />. Then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667041.png" />-bi-invariant function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667042.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667043.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667044.png" /> is called a spherical function. Sometimes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667045.png" /> is called a zonal spherical function, while the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667046.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667047.png" />) are also called spherical functions. Some authors call <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667048.png" /> an elementary spherical function, while all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667049.png" />-bi-invariant functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667050.png" /> are called spherical functions.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667051.png" /> is a Gel'fand pair if, for all irreducible unitary representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667052.png" />, the subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667053.png" />-fixed vectors in the representation space has dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667054.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667055.png" />. This is equivalent to the commutativity of the convolution algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667056.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667057.png" />-bi-invariant continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667058.png" /> with compact support. Now spherical functions are more generally defined as solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667059.png" />, not identically zero, of the functional equation
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667060.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
\phi ( x) \phi ( y)  = \int\limits _ { K } \phi ( xky) dk,\  x, y \in G,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667061.png" /> is the normalized Haar measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667062.png" />. These solutions include the spherical functions associated with irreducible unitary representations. Other solutions may be associated with irreducible non-unitary representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667063.png" />. The characters of the comutative algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667064.png" /> are precisely the mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667065.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667066.png" /> is Haar measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667067.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667068.png" /> is a solution of (*).
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667069.png" /> is, moreover, a connected Lie group, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667070.png" /> is a Gel'fand pair if and only if the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667071.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667072.png" />-invariant differential operators on the homogeneous space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667073.png" /> is commutative. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667074.png" /> is a solution of (*) if and only if it is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667075.png" />-bi-invariant, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667076.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667077.png" />, and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667078.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667079.png" /> is a joint eigenfunction of the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667080.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667081.png" /> is a connected real semi-simple LIe group and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667082.png" /> is a maximal compact subgroup, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667083.png" /> is a Gel'fand pair, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667084.png" /> is a Riemannian symmetric space, and much information is available about <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086670/s08667085.png" /> and the sperical functions.
+
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====
 
====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>
 
<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>

Latest revision as of 08:22, 6 June 2020


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
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