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

Giacomo Gigante¹

¹ Dipartimento di Ingegneria dell'Informazione e Metodi Matematici Università di Bergamo Viale Marconi 5, 24044 Dalmine (BG), Italy

Colloquium Mathematicum, Tome 119 (2010) no. 2, pp. 237-254

Résumé

Let $\{A_{k}\}_{k=0}^{+\infty}$ be a sequence of arbitrary complex numbers, let $\alpha, \beta>-1$, let $\{P_{n}^{\alpha, \beta}\}_{n=0}^{+\infty}$ be the Jacobi polynomials and define the functions $$\eqalign{ H_{n}(\alpha, z) =\sum_{m=n}^{+\infty}\frac{A_{m}z^{m}}{{\mit\Gamma}( \alpha+n+m+1) ( m-n) !},\cr G( \alpha, \beta, x, y) =\sum_{r,s=0}^{+\infty}\frac{A_{r+s}x^{r}y^{s}}{{\mit\Gamma}( \alpha+r+1) {\mit\Gamma}( \beta+s+1) r!s!}.\cr}$$ Then, for any non-negative integer $n$, \begin{multline*} \int_{0}^{{\pi}/{2}}G( \alpha, \beta, x^{2}\sin^{2}\phi , y^{2}\cos^{2}\phi) P_{n}^{\alpha, \beta}( \cos2\phi) \sin^{2\alpha+1}\phi\cos^{2\beta+1}\phi \,d \\ =\frac{1}{2}H_{n}( \alpha+\beta+1, x^{2}+y^{2}) P_{n}^{\alpha, \beta} \left( \frac{y^{2}-x^{2}}{y^{2}+x^{2}}\right) .\end{multline*} When $A_{k}=( -1/4) ^{k}$, 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 one to write all these type of formulas for specific special functions, like Gegenbauer, Jacobi and Laguerre polynomials, Jacobi functions, or hypergeometric functions.

Détail

DOI : 10.4064/cm119-2-6

Keywords: infty sequence arbitrary complex numbers alpha beta alpha beta infty jacobi polynomials define functions eqalign alpha sum infty frac mit gamma alpha m n alpha beta sum infty frac mit gamma alpha mit gamma beta non negative integer begin multline* int alpha beta sin phi cos phi alpha beta cos phi sin alpha phi cos beta phi frac alpha beta alpha beta frac x right end multline* formula reduces batemans expansion bessel functions particular values obtains generalizations several formulas already known bessel functions sonines first second finite integrals certain neumann series expansions particular choices infty allow write these type formulas specific special functions gegenbauer jacobi laguerre polynomials jacobi functions hypergeometric functions