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A kind of orthogonal polynomials and related identities. (arXiv:1606.08327v3 [math.NT] UPDATED)
来源于:arXiv
In this paper we introduce the polynomials $\{d_n^{(r)}(x)\}$ and
$\{D_n^{(r)}(x)\}$ given by
$d_n^{(r)}(x)=\sum_{k=0}^n\binom{x+r+k}k\binom{x-r}{n-k} \ (n\ge 0)$,
$D_0^{(r)}(x)=1,\ D_1^{(r)}(x)=x$ and
$D_{n+1}^{(r)}(x)=xD_n^{(r)}(x)-n(n+2r)D_{n-1}^{(r)}(x)\ (n\ge 1).$ We show
that $\{D_n^{(r)}(x)\}$ are orthogonal polynomials for $r>-\frac 12$, and
establish many identities for $\{d_n^{(r)}(x)\}$ and $\{D_n^{(r)}(x)\}$,
especially obtain a formula for $d_n^{(r)}(x)^2$ and the linearization formulas
for $d_m^{(r)}(x)d_n^{(r)}(x)$ and $D_m^{(r)}(x)D_n^{(r)}(x)$. As an
application we extend recent work of Sun and Guo. 查看全文>>