Bender–Dunne polynomials

In mathematics, Bender–Dunne polynomials are a two-parameter family of sequences of orthogonal polynomials studied by Bender and Dunne (1988, 1996). They may be defined by the recursion:

${\displaystyle P_{0}(x)=1}$,

${\displaystyle P_{1}(x)=x}$ ,

and for $n>1$:

${\displaystyle P_{n}(x)=xP_{n-1}(x)+16(n-1)(n-J-1)(n+2s-2)P_{n-2}(x)}$

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

References

• Bender, Carl M.; Dunne, Gerald V. (1988), "Polynomials and operator orderings", Journal of Mathematical Physics, 29 (8): 1727–1731, doi:10.1063/1.527869, ISSN 0022-2488, MR 955168
• Bender, Carl M.; Dunne, Gerald V. (1996), "Quasi-exactly solvable systems and orthogonal polynomials", Journal of Mathematical Physics, 37 (1): 6–11, doi:10.1063/1.531373, ISSN 0022-2488, MR 1370155