Ahmed, H. (2017). Recurrences and explicit formulae for the expansion and connection coefficients in series of the product of two classical discrete orthogonal polynomials. Bulletin of the Iranian Mathematical Society, 43(7), 2585-2615.

H.M. Ahmed. "Recurrences and explicit formulae for the expansion and connection coefficients in series of the product of two classical discrete orthogonal polynomials". Bulletin of the Iranian Mathematical Society, 43, 7, 2017, 2585-2615.

Ahmed, H. (2017). 'Recurrences and explicit formulae for the expansion and connection coefficients in series of the product of two classical discrete orthogonal polynomials', Bulletin of the Iranian Mathematical Society, 43(7), pp. 2585-2615.

Ahmed, H. Recurrences and explicit formulae for the expansion and connection coefficients in series of the product of two classical discrete orthogonal polynomials. Bulletin of the Iranian Mathematical Society, 2017; 43(7): 2585-2615.

Recurrences and explicit formulae for the expansion and connection coefficients in series of the product of two classical discrete orthogonal polynomials

^{1}Department of Mathematics, Faculty of Industrial Education, Helwan University, Cairo-Egypt

^{2}Department of Mathematics, Faculty of Sciences, Saqraa University, Shaqraa-KSA.

Receive Date: 11 April 2017,
Revise Date: 04 January 2018,
Accept Date: 05 January 2018

Abstract

Suppose that for an arbitrary function $f(x,y)$ of two discrete variables, we have the formal expansions. [f(x,y)=sumlimits_{m,n=0}^{infty }a_{m,n},P_{m}(x)P_{n}(y),] $$ x^{m}P_{j}(x)=\sum\limits_{n=0}^{2m}a_{m,\,n}(j)P_{j+m-n}(x),$$ we find the coefficients $b_{i,j}^{(p,q,\ell ,\,r)}$ in the expansion $$ x^{\ell }y^{r}\,\nabla _{x}^{p}\nabla _{y}^{q}\,f(x,y)=x^{\ell }y^{r}f^{(p,q)}(x,y) =\sum\limits_{m,n=0}^{\infty }a_{m,n}^{(p,q)}\,P_{m}(x)P_{n}(y),\,\,a_{m,n}^{(0,0)}=a_{m,n},$$ We give applications of these results in solving partial difference equations with varying polynomial coefficients, by reducing them to recurrence relations (difference equations) in the expansion coefficients of the solution.

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