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The [[Integral transform|integral transform]]  $  T \{ F( t) \} $
of a function  $  F( t) $,

$$ 
T \{ F ( t) \}  = \ 
\int\limits _ { - } 1 ^ { + }  1
( 1 - t  ^ {2} ) ^ {\rho - 1/2 }
C _ {n}  ^  \rho  ( t) F ( t)  dt  =  f _ {n} ^ { \rho } ,
$$

$$ 
\rho  >  - 
\frac{1}{2}
 ,\  n  =  0, 1 , .  .  . .
$$

Here  $  C _ {n}  ^  \rho  $
are the [[Gegenbauer polynomials|Gegenbauer polynomials]]. If a function can be expanded into a generalized Fourier series by Gegenbauer polynomials, the following inversion formula is valid:

$$ 
F ( t)  =  \sum _ {n = 0 } ^  \infty  

\frac{n! ( n + \rho ) \Gamma  ^ {2} ( \rho ) 2 ^ {2 \rho - 1 } }{\pi \Gamma ( n + 2 \rho ) }

C _ {n}  ^  \rho  ( t) f _ {n} ^ { \rho } ,\ \ 
- 1 < t < 1.
$$

The Gegenbauer transform reduces the differentiation operation

$$ 
R [ F ( t)]  = \ 
( 1 - t  ^ {2} ) F ^ { \prime\prime } - ( 2 \rho - 1) tF ^ { \prime\prime }
$$

to the algebraic operation

$$ 
T \{ R [ F ( t)] \}  =  - n ( n + 2 \rho ) f _ {n} ^ { \rho } .
$$

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.A. Ditkin,   A.P. Prudnikov,   "Operational calculus"  ''Progress in Math.'' , '''1'''  (1968)  pp. 1–75  ''Itogi Nauk. Ser. Mat. Anal. 1966''  (1967)  pp. 7–82</TD></TR></table>

====Comments====
For any system of orthogonal polynomials one can formally consider a transform pair as above, cf.

and

in [[Fourier series in orthogonal polynomials|Fourier series in orthogonal polynomials]]. The Gegenbauer transform (and, more generally, the [[Jacobi transform|Jacobi transform]]) has been considered for arguments  $  n $
which are arbitrarily complex. Then inversion formulas exist in the form of integrals and there is a relationship with sampling theory, cf. [[#References|[a1]]], [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.L. Butzer,   R.L. Stens,   M. Wehrens,   "The continuous Legendre transform, its inverse transform, and applications,"  ''Internat. J. Math. Sci.'' , '''3'''  (1980)  pp. 47–67</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  T.H. Koornwinder,   G.G. Walter,   "The finite continuous Jacobi transform and its inverse"  ''J. Approx. Theory''  (To appear)</TD></TR></table>