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

Tom 119 / 2010

Giacomo Gigante Colloquium Mathematicum 119 (2010), 237-254 MSC: 33C45; 42C05. DOI: 10.4064/cm119-2-6

Streszczenie

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.

Autorzy

  • Giacomo GiganteDipartimento di Ingegneria dell'Informazione e Metodi Matematici
    Università di Bergamo
    Viale Marconi 5, 24044 Dalmine (BG), Italy
    e-mail

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