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 Dunne (1988, 1996). They may be defined by the recursion:

[math]\displaystyle{ P_0(x) = 1 }[/math],

[math]\displaystyle{ P_{1}(x) = x }[/math] ,

and for [math]\displaystyle{ n \gt 1 }[/math]:

[math]\displaystyle{ P_n(x) = x P_{n-1}(x) + 16 (n-1) (n-J-1) (n + 2 s -2) P_{n-2}(x) }[/math]

where [math]\displaystyle{ J }[/math] and [math]\displaystyle{ s }[/math] 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, Bibcode: 1988JMP....29.1727B 
• 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, Bibcode: 1996JMP....37....6B