Tricomi–Carlitz polynomials

In mathematics, the Tricomi–Carlitz polynomials or (Carlitz–) Karlin–McGregor polynomials are polynomials studied by Tricomi (1951),^[1] Carlitz (1958),^[2] and Karlin and McGregor (1959),^[3] related to random walks on the positive integers.

They are given in terms of Laguerre polynomials by

${\displaystyle \ell _{n}(x)=(-1)^{n}L_{n}^{(x-n)}(x).}$

They are special cases of the Chihara–Ismail polynomials.

References

1. Tricomi, Francesco G. (December 1951). "A class of non-orthogonal polynomials related to those of Laguerre". Journal d'Analyse Mathématique. 1 (1): 209–231. doi:10.1007/BF02790089. ISSN 0021-7670.
2. Carlitz, Leonard (1958), "On some polynomials of Tricomi", Boll. Un. Mat. Ital. (3), 13: 58–64, MR 0103303
3. Karlin, Samuel; McGregor, James (1959-03-01). "Random walks". Illinois Journal of Mathematics. 3 (1). doi:10.1215/ijm/1255454999. ISSN 0019-2082.