Orthogonal polynomials for geometrically discounted series

Question

What class of polynomials satisfy the orthogonality condition:

$$\sum_{k=0}^\infty{\rho^k p_i(k)p_j(k)}=0 \Leftrightarrow i\ne j$$

for $|\rho|<1$?

Answer 1

These polynomials are given by the following formula $$p_n(x)=\sum_{j=0}^{n}(-\rho)^j\binom{n}{j}\binom{x+j}{n}$$ (up to a factor not depending on $x$). Indeed, \begin{align} \sum_{k=0}^{\infty}\rho^k k^m p_n(k)&=\sum_{j=0}^{n}(-\rho)^j\binom{n}{j}\sum_{k=\color{red}{n-j}}^{\infty}\rho^k k^m\binom{k+j}{n}\\&=\sum_{j=0}^{n}(-\rho)^j\binom{n}{j}\sum_{k=0}^{\infty}\rho^{k+n-j}(k+n-j)^m\binom{k+n}{n}\\&=\rho^n\sum_{j=0}^{n}(-1)^j\binom{n}{j}\sum_{k=0}^{\infty}\rho^k(k+n-j)^m\binom{k+n}{n}\\&=\rho^n\sum_{k=0}^{\infty}\binom{k+n}{n}\rho^k\sum_{j=0}^{n}(-1)^{n-j}\binom{n}{j}(k+j)^m. \end{align} The inner sum is $0$ if $m<n$ (and is $n!$ if $m=n$). Thus, $$\sum_{k=0}^{\infty}\rho^k p_m(k) p_n(k)=\begin{cases}\rho^n/(1-\rho),& m=n\\ \hfill 0,\hfill& m\neq n\\ \end{cases}.$$