Bender–Dunne polynomials explained

In mathematics, Bender–Dunne polynomials are a two-parameter family of sequences of orthogonal polynomials studied by Carl M. Bender and Gerald V. Dunne.^{[1] [2]} 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.

Notes and References

1. Bender . Carl M. . Dunne . Gerald V. . Polynomials and operator orderings . 10.1063/1.527869 . 955168 . 1988 . . 0022-2488 . 29 . 8 . 1727–1731. 1988JMP....29.1727B.
2. Bender . Carl M. . Dunne . Gerald V. . Quasi-exactly solvable systems and orthogonal polynomials . 10.1063/1.531373 . 1370155 . 1996 . . 0022-2488 . 37 . 1 . 6–11. hep-th/9511138 . 1996JMP....37....6B . 28967621.