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Let ${A_{k}}_{k=0}^{+infty}$ be a sequence of arbitrary complex numbers, let $alpha,, eta>-1$, let ${P_{n}^{alpha,, eta}}_{n=0}^{+infty}$ be the Jacobi polynomials and define the functions%  egin{align*} H_{n}left( alpha,,z ight) & =sum_{m=n}^{+infty}frac{A_{m}z^{m}% }{Gammaleft( alpha+n+m+1 ight) left( m-n ight) !}\ Gleft( alpha,, eta,,x,,y ight) & =sum_{r,s=0}^{+infty}% frac{A_{r+s}x^{r}y^{s}}{Gammaleft( alpha+r+1 ight) Gammaleft(  eta+s+1 ight) r!s!}. end{align*} Then, for any non negative integer $n$,%  egin{multline*} int_{0}^{frac{pi}{2}}Gleft( alpha,, eta,,x^{2}sin^{2}phi ,,y^{2}cos^{2}phi ight) P_{n}^{alpha,, eta}left( cos2phi ight) ,sin^{2alpha+1}phicos^{2 eta+1}phi,dphi\ =frac{1}{2}H_{n}left( alpha+ eta+1,,x^{2}+y^{2} ight) P_{n}% ^{alpha,, eta}left( frac{y^{2}-x^{2}}{y^{2}+x^{2}} ight) . end{multline*}   When $A_{k}=left( -1/4 ight) ^{k}$, this formula reduces to Bateman's expansion for Bessel functions. For particular values of $y$ and $n$ one obtains generalizations of several formulas already known for Bessel functions, like Sonine's first and second finite integrals and certain Neumann series expansions. Particular choices of ${A_{k}}_{k=0}^{+infty}$ allow to write all these type of formulas for specific special functions, like Gegenbauer, Jacobi and Laguerre polynomials, Jacobi functions, or hypergeometric functions.

A generalization of Bateman's expansion and finite integrals of Sonine's and Feldheim's type