论文标题
一类高几何多项式的新线性反演公式
A New Linear Inversion Formula for a class of Hypergeometric polynomials
论文作者
论文摘要
给定复杂的参数$ x $，$ν$，$α$，$β$和$γ\ notin -\ mathbb {n} $，请考虑无限的下三角矩阵$ \ mathbf {a}（x，x，x，n n n a};α，β，β，γ）$带有element \ displayStyle（-1）^k \ binom {n+α} {k+α} \ cdot f（k-n， - （β+n）ν; - （β+n）n; - （γ+n）; x）$ for $ 1 \ leqslant k \ leqslant k \ leqslant n $，根据cd cd cd cd cd $ cd; $ n \ in \ mathbb {n}^*$。在陈述了无限生成函数的无限矩阵反转的一般标准之后，我们证明了逆矩阵$ \ mathbf {b} {b}（x，x，v;α，β，β，γ，γ）= \ mathbf {a}（a}（x，x，x，x，x;α，β，β，β，γ） b_ {n，k}（x，v;α，β，γ）=＆\; \ displayStyle（-1）^k \ binom {n+α} {k+α} \; \ cdot \ nonumber \\＆\; \ biggl [\; \ frac {γ+k} {β+k} \，f（k-n，（β+k）ν;γ+k; x）\; + \ nonumber \\＆\; \; \; \ frac {β-γ} {β+k} \，f（k-n，（β+k）ν; 1+γ+k; x）\; \ biggr] \ nonumber \ end {align}对于$ 1 \ leqslant k \ leqslant n $，从而提供了新的线性反转公式。相关序列$ s $和$ t $的生成函数的功能关系，即$ t = \ mathbf {a}（x，x，x，x，n;α，β，γ）\，s \ longleftrightArol s = \ mathbf {b}（b}（b}（x，x，x，x，x，x;α，α，β，γ）\，也提供了。
Given complex parameters $x$, $ν$, $α$, $β$ and $γ\notin -\mathbb{N}$, consider the infinite lower triangular matrix $\mathbf{A}(x,ν;α, β,γ)$ with elements $$ A_{n,k}(x,ν;α,β,γ) = \displaystyle (-1)^k\binom{n+α}{k+α} \cdot F(k-n,-(β+n)ν;-(γ+n);x) $$ for $1 \leqslant k \leqslant n$, depending on the Hypergeometric polynomials $F(-n,\cdot;\cdot;x)$, $n \in \mathbb{N}^*$. After stating a general criterion for the inversion of infinite matrices in terms of associated generating functions, we prove that the inverse matrix $\mathbf{B}(x,ν;α, β,γ) = \mathbf{A}(x,ν;α, β,γ)^{-1}$ is given by \begin{align} B_{n,k}(x,ν;α, β,γ) = & \; \displaystyle (-1)^k\binom{n+α}{k+α} \; \cdot \nonumber \\ & \; \biggl [ \; \frac{γ+k}{β+k} \, F(k-n,(β+k)ν;γ+k;x) \; + \nonumber \\ & \; \; \; \frac{β-γ}{β+k} \, F(k-n,(β+k)ν;1+γ+k;x) \; \biggr ] \nonumber \end{align} for $1 \leqslant k \leqslant n$, thus providing a new class of linear inversion formulas. Functional relations for the generating functions of related sequences $S$ and $T$, that is, $T = \mathbf{A}(x,ν;α, β,γ) \, S \Longleftrightarrow S = \mathbf{B}(x,ν;α, β,γ) \, T$, are also provided.
