This started when I was examining certain families of unimodal polynomials, i.e. $\sum_{k=0}^n a_kx^k$ where $a_0\le a_1\le\cdots \le a_k\ge\cdots \ge a_n$. (Notation: in the follo …

I would like to know if there is a proof for the identity used in the superconformal index of 4d ${\cal N}=2$ gauge theory. In the paper by Rastelli el al, it was discovered that E …

Let ${p_n}(x) = x{p_{n - 1}}(x) - {a_{n - 2}}{p_{n - 2}}(x)$ for some numbers ${a_n}$ with initial values ${p_{ - 1}}(x) = 0$ and ${p_0}(x) = 1.$ By Favard’s theorem about ortho …

For given $n,\ell\in\mathbb N_0$, I am interested in studying the following recursion relation for some $\mu\in\mathbb R$: $$\sqrt{-1} \tfrac{j(\ell-j+1)(n-j+1)(n+j+1)}{2(2j-1)(2j+ …

Jacobi polynomials, denoted by $J^{(\alpha,\beta)}_n$, on $[-1,1]$ satisfy a three term recurrence $$ J_{n+1}^{(\alpha,\beta)}(x) = (A_n+B_nx)J^{(\alpha,\beta)}_n + C_nJ_{n-1}^{(\ …

I have a minor result which I'm sure has come up somewhere before but I can't seem to find it. Consider a confluent hypergeometric function of the form $$\newcommand{\ff}{{}_1F_1 …

Let $\mu$ be a measure on $\def\r{\mathbb{R}}\r^n$, $1\le n \le \infty$. Given a (finite) multi-index $\bar{i} = (i_1, i_2, \ldots)$, one can define the moment $$ m_{\bar i} = \in …

Where can I find generating functions for orthogonal polynomials in two variables? Lebedev's book (Special Functions and their Applications, Dover, 1972) gives a closed form for $$ …

I am looking for an othogonal basis of functions over the interval $[0,1]$ with weight function $(1-x^2)^{\alpha-1/2}$. Gegenbauer polynomials are frustratingly close to what I nee …

I'd like to know the asymptotic behaviour as $N\to\infty$ of the following sum $$ Z_N(x) := 2^{-N/2} \sum_{k=0}^{N/2} \frac{N!}{k! (N-2k)!} (N-1)^{-k} (\sqrt{2} x)^{N-2k} $$ in o …

Let $P_{k}^{\alpha}(q)$ be generalized Laguerre polynomials defined as $$P_{k}^{\alpha}(q)=\sum_{r=0}^k \tbinom{k}{r}(-1)^r\frac{\Gamma(k+\alpha+1)}{\Gamma(k+\alpha-r+1)}q^{k-r},$ …

Let $ L_{d}^{(1)}(x)$ denote the generalized Laguerre polynomial of degree $d$ and order $\alpha=1$. Clearly, since all the roots $r_1,\dots,r_d$ of $L_{d}^{(1)}$ are simple, there …

Let $\lbrace P_n(z)\rbrace_{n\in\mathbb N_0}$ be a family of polynomials defined by a generating function $g(t,z)=\sum\limits_{n=0}^\infty P_n(z)t^n$ or by a contour integral $P_n …

I have a collection of orthogonal polynomials in infinitely commuting variables $x_1, x_2, x_3, \ldots$. I think they must be well known (perhaps Schur or Hermite polynomials or s …

Hi! I've been using the Christoffel-Darboux identity for the Hermite polynomials, $$\sum_{k=0}^n\frac{H_k(x)H_k(y)}{2^k k!}=\frac{1}{2^n n!}\frac{H_{n+1}(x)H_n(y)-H_n(x)H_{n+1}(y)} …