论文标题
晶格上正交多项式的表征
Characterization of Orthogonal Polynomials on lattices
论文作者
论文摘要
我们考虑正交多项式的两个序列$（p_n）_ {n \ geq 0} $和$（q_n）_ {n \ geq 0} $，这样的$ \ sum_ {j = 1} ^{m} ^{m} a________ { p_ {k+n-j}（z）= \ sum_ {j = 1} ^{n} b_ {j，n} \ mathrm {d} _x ^{m} q_ {m+n-j}（m+n-j}（z）（z）\;，$ k，m，m，m，m，m，m，m，n \ n} $ b_ {j，n} $是复数的序列，$$ 2 \ mathrm {s} _xf（x（s））=（\ triangle +2 \，\ Mathrm {i}）f（z）f（z），~~ \ m mathrm {d} x（s-1/2）} f（z），$$ $ z = x（s-1/2）$，$ \ mathrm {i} $是身份操作员，$ x $定义了一个晶格和$ \ triangle f（s）= f（s+1）-f（s+1）-f（s）-f（s）$。我们表明，在某些自然条件下，两者都涉及正交多项式序列$（p_n）_ {n \ geq 0} $和$（q_n）_ {n \ geq 0} $，每当$ k = m $时，都是半典型的。仔细研究了一些特殊情况，我们表征了连续的双Hahn和Wilson多项式用于二次晶格。
We consider two sequences of orthogonal polynomials $(P_n)_{n\geq 0}$ and $(Q_n)_{n\geq 0}$ such that $$ \sum_{j=1} ^{M} a_{j,n}\mathrm{S}_x\mathrm{D}_x ^k P_{k+n-j} (z)=\sum_{j=1} ^{N} b_{j,n}\mathrm{D}_x ^{m} Q_{m+n-j} (z)\;, $$ with $k,m,M,N \in \mathbb{N}$, $a_{j,n}$ and $b_{j,n}$ are sequences of complex numbers, $$2\mathrm{S}_xf(x(s))=(\triangle +2\,\mathrm{I})f(z),~~ \mathrm{D}_xf(x(s))=\frac{\triangle}{\triangle x(s-1/2)}f(z),$$ $z=x(s-1/2)$, $\mathrm{I}$ is the identity operator, $x$ defines a lattice, and $\triangle f(s)=f(s+1)-f(s)$. We show that under some natural conditions, both involved orthogonal polynomials sequences $(P_n)_{n\geq 0}$ and $(Q_n)_{n\geq 0}$ are semiclassical whenever $k=m$. Some particular cases are studied closely where we characterize the continuous dual Hahn and Wilson polynomials for quadratic lattices.
