Bender–Dunne polynomials

In mathematics, Bender–Dunne polynomials are a two-parameter family of sequences of orthogonal polynomials studied by Carl M. Bender and Gerald V. Dunne.{{cite journal |last1=Bender |first1=Carl M. |last2=Dunne |first2=Gerald V. |title=Polynomials and operator orderings |doi=10.1063/1.527869 |mr=955168 |year=1988 |journal=Journal of Mathematical Physics |issn=0022-2488 |volume=29 |issue=8 |pages=1727–1731| bibcode=1988JMP....29.1727B}}{{cite journal |last1=Bender |first1=Carl M. |last2=Dunne |first2=Gerald V. |title=Quasi-exactly solvable systems and orthogonal polynomials |doi=10.1063/1.531373 |mr=1370155 |year=1996 |journal=Journal of Mathematical Physics |issn=0022-2488 |volume=37 |issue=1 |pages=6–11| arxiv=hep-th/9511138 |bibcode=1996JMP....37....6B |s2cid=28967621}} They may be defined by the recursion:

: $P\_0(x)\; =\; 1$,

: $P\_\{1\}(x)\; =\; x$ ,

and for $n\; >\; 1$:

: $P\_n(x)\; =\; x\; P\_\{n-1\}(x)\; +\; 16\; (n-1)\; (n-J-1)\; (n\; +\; 2\; s\; -2)\; P\_\{n-2\}(x)$

where $J$ and $s$ are arbitrary parameters.