Sobolev orthogonal polynomials generated by Meixner polynomials

I. I. Sharapudinov ; Z. D. Gadzhieva

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 16 (2016) no. 3, pp. 310-321

Résumé

The problem of constructing Sobolev orthogonal polynomials $m _{r,n}^{\alpha}(x,q)$ $(n=0,1,\ldots)$, generated by classical Meixner's polynomials is considered. They can by defined using the following equalities $m_{r,k}^{\alpha}(x,q)={x^{[k]}\over k!}$, $x^{[k]}=x(x-1)\cdots(x-k+1)$, $k=0,1,\ldots,r-1$, $m_{r,k+r}^{\alpha}(x,q)=\frac{1}{(r-1)!}\sum\limits_{t=0}^{x-r}(x-1-t)^{[r-1]}m_{k}^{\alpha}(t,q)$, where $m_{k}^{\alpha}(t,q)$ denote Meixner's polynomial of degree $k$, orthonormal on $\Omega=\{0,1,\ldots\}$ with weight $\rho(x)=q^x\frac{\Gamma(x+\alpha+1)}{\Gamma(x+1)}(1-q)^{\alpha+1}$. Polynomials $m _{r,n}^{\alpha}(x,q)$, $(n=0,1,\ldots)$ are orthonormal on $\Omega=\{0,1,\ldots\}$ with respect to the inner product $$ \langle m_{r,n}^{\alpha},m_{r,m}^{\alpha}\rangle= \sum\limits_{k=0}^{r-1}\Delta^km_{r,n}^{\alpha}(0,q)\Delta^km_{r,m}^{\alpha}(0,q)+ \sum\limits_{j=0}^{\infty}\Delta^rm_{r,n}^{\alpha}(j,q)\Delta^r m_{r,m}^{\alpha}(j,q)\rho(j). $$ For $m_{r,n}^{\alpha}(x,q)$ we obtain the explicit formula that contains the Мeixner polynomial $M_{n}^{\alpha-r}(x,q)$: $$ m_{r,k+r}^{\alpha}(x,q)=\big(\frac{q}{q-1}\big)^r\left\{h_{k}^{\alpha}(q)\right\}^{-1/2} \left[M_{k+r}^{\alpha-r}(x,q)-\sum\limits_{\nu=0}^{r-1}\frac{A_{r,k,\nu}x^{[\nu]}}{\nu!}\right], k=0,1,\ldots, $$ where $A_{r,k,\nu}=\Big({q-1\over q}\Big)^\nu \frac{\Gamma(k+\alpha+1)}{(k+r-\nu)!\Gamma(\nu-r+\alpha+1)}$, $M_n^\alpha(x,q)=\frac{\Gamma (n+\alpha+1)}{n!} \sum_{k=0}^n{n^{[k]}x^{[k]}\over \Gamma (k+\alpha+1)k!}\left(1-{1\over q}\right)^k$, $h_n^\alpha(q)= {n+\alpha\choose n}q^{-n}\Gamma(\alpha+1)$.