Generating relations of Tricomi and Hermite-Tricomi functions using Lie algebra representation
Let \(C_n(x)=\sum_{k=0}^\infty(-1)^kx^k/(k!(n+k)!)\) be a Tricomi function and \[ H_n(x,y)=n!\sum_{k=0}^{[n/2]}y^k x^{n-2k}/(k!(n-2k)!) \] be a Hermite polynomial. 2-index 3-variable extension of Tricomi function (2I3VTF) \[ C_{m,n}(x,y,z)= \sum_{k=0}^\infty C_{m+k}(x) C_{n+k}(y)z^k/k! \] and 2-index 4-variable 1-parameter Hermite-Tricomi function (2I4V1PHTF) [see \textit{G. Dattoli}, J. Math. Anal. Appl. 269, No. 2, 716--725 (2002; Zbl 1073.33008)] \[ {}_H C_{m,n}(x,y;z,w/\xi)=\sum_{r,s=0}^\infty \frac{(-1)^{r+s}H_{r,s}(x,y;z,w/\xi)}{r!s!(m+r)!(n+s)!}, \] where \(H_{r,s}(x,y;z,w/\xi)\) is a Hermite polynomial \[ H_{m,n}(x,y;z,w/\xi)=m!n!\sum_{r=0}^{\min(m,n)}\frac{\xi^r H_{m-r}(x,y)H_{n-r}(z,w)}{r!(m-r)!(n-r)!} \] are considered. Generating relations of \(C_{m,n}(x,y,z)\) and \({}_H C_{m,n}(x,y;z,w/\xi)\) by using Lie-algebraic methods involving various forms of Tricomi, Hermite-Tricomi and Bessel functions are obtained. Some properties of \(C_{m,n}\) and \({}_H C_{m,n}\) are also given.
scientific article; zbMATH DE number 5318698
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