Tricomi–Carlitz polynomials

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

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

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