Spider Flower – Cleome

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

They are given in terms of Laguerre polynomials by

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

They are special cases of the Chihara–Ismail polynomials.

References

Carlitz, Leonard (1958), "On some polynomials of Tricomi", Boll. Un. Mat. Ital. (3), 13: 58–64, MR 0103303
Karlin, Samuel; McGregor, James (1959), "Random walks", Illinois Journal of Mathematics, 3: 66–81, doi:10.1215/ijm/1255454999, ISSN 0019-2082, MR 0100927
Tricomi, Francesco G. (1951), "A class of non-orthogonal polynomials related to those of Laguerre", Journal d'Analyse Mathématique, 1: 209–231, doi:10.1007/BF02790089, ISSN 0021-7670, MR 0051351