A generalization of binary Krawtchouk polynomials

Question

I am looking for orthogonal polynomials $P_{d,m,n}$ that have their values at integers $i$ specified by the following generating function

$(1-z)^i (1+z+z^2+ \ldots + z^d)^{n-i} = \sum_{m=0}^{i+d(n-i)} P_{d,m,n}(i) \cdot z^m$

That is,

$P_{d,m,n}(i) = \sum_{j=0}^{m} \big[ (-1)^j \binom{i}{j} \sum_{k_0+\cdots+k_d = n-i, \mbox{ } 1k_1+2k_2+\cdots+dk_d = m-j} \binom{n-i}{k_0,k_1\ldots,k_d} \big]$

The polynomials $P_{d,m,n}$ can be seen as a generalization of binary Krawtchouk Polynomials $K_{m,n,2}$ (see e.g. https://arxiv.org/abs/1101.1798) which satisfy

$(1-z)^i (1+z)^{n-i} = \sum_{m=0}^{n} K_{m,n,2}(i) \cdot z^m$

and

$K_{m,n,2}(i) = \sum_{j=0}^{m} (-1)^j \binom{i}{j}\binom{n-i}{m-j}$

I looked into multivariate Krawtchouk and Meixner polynomials but wasn't able to derive the above generating function from those. Can someone point me to existing literature or give me a hint? Even the family $P_{2,m,n}$, i.e. $d=2$, is of particular interest to me.

Answer 1

From the formula for $P_{d,m,n}(i)$ it follows that $$P_{d,m,n}(i) = \sum_j (-1)^j \binom{i}{j} \sum_{k_0\geq 0} \binom{n-i}{k_0} \hat{B}_{m-j,n-i-k_0}(\underbrace{1,1,\ldots,1}_{d\ \text{ones}},0,0,\ldots),$$ where $\hat{B}_{m-j,n-i-k_0}()$ is the partial ordinary Bell polynomials. Recalling the generating function for ordinary Bell polynomials, one can further obtain that $P_{d,m,n}(i)$ equals the coefficient of $t^m$ in $$(1-t)^{2i-n}(1-t^{d+1})^{n-i}.$$ This further implies the following explicit formula: $$P_{d,m,n}(i) = \sum_{k=0}^{\lfloor m/(d+1)\rfloor} (-1)^{m-kd} \binom{n-i}{k} \binom{2i-n}{m-k(d+1)}.$$ It follows that $P_{d,m,n}(i)$ is a polynomial of degree $m$ in variable $i$.