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

Document Type : Research Paper

Author

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

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

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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References

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Bulletin of the Iranian Mathematical Society
Volume 43, Issue 7
December 2017
Pages 2585-2615

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History

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

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