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Let ${A_k}_{k=0}^{+\infty }$ be a sequence of arbitrary complex numbers, let α,β > -1, let Pₙα,βn=0+∞$betheJacobipolynomialsanddefinethefunctions$ $Hₙ(\alpha ,z)={\sum }_{m=n}^{+\infty }\left(A_m{z}^m\right)/(\Gamma (\alpha +n+m+1)(m-n)!)$, $G(\alpha ,\beta ,x,y)={\sum }_{r,s=0}^{+\infty }\left(A_{r+s}{x}^r{y}^s\right)/(\Gamma (\alpha +r+1)\Gamma (\beta +s+1)r!s!)$. Then, for any non-negative integer n, ${\int }_0^{\pi /2}G(\alpha ,\beta ,x²sin²\varphi ,y²cos²\varphi )P{ₙ}^{\alpha ,\beta }\left(cos²\varphi \right)si{n}^{2\alpha +1}\varphi co{s}^{2\beta +1}\varphi d=1/2Hₙ(\alpha +\beta +1,x²+y²)P{ₙ}^{\alpha ,\beta }((y²-x²)/(y²+x²))$. 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...

A transference theorem for convolution operators is proved for certain families of one-dimensional hypergroups.