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Q-analogues of the Fibo-Stirling numbers by Bach, Quang T.; Paudyal, Roshil; Remmel, Jeffrey B. is a scholarly article available to read on EtoBox.

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Let $F_n$ denote the $n^{th}$ Fibonacci number relative to the initial conditions $F_0=0$ and $F_1=1$. Bach, Paudyal, and Remmel introduced Fibonacci analogues of the Stirling numbers called Fibo-Stirling numbers of the first and second kind. These numbers serve as the connection coefficients between the Fibo-falling factorial basis $\{(x)_{\downarrow_{F,n}}:n \geq 0\}$ and the Fibo-rising factorial basis $\{(x)_{\uparrow_{F,n}}:n \geq 0\}$ which are defined by $(x)_{\downarrow_{F,0}} = (x)_{\uparrow_{F,0}} = 1$ and for $k \geq 1$, $(x)_{\downarrow_{F,k}} = x(x-F_1) \cdots (x-F_{k-1})$ and $(x)_{\uparrow_{F,k}} = x(x+F_1) \cdots (x+F_{k-1})$. We gave a general rook theory model which allowed us to give combinatorial interpretations of the Fibo-Stirling numbers of the first and second kind. There are two natural $q$-analogues of the falling and rising Fibo-factorial basis. That is, let $[x]_q = \frac{q^x-1}{q-1}$. Then we let $[x]_{\downarrow_{q,F,0}} = \overline{[x]}_{\downarrow_{q,F,0}} = [x]_{\uparrow_{q,F,0}} = \overline{[x]}_{\uparrow_{q,F,0}}=1$ and, for $k > 0$, we let $[x]_{\downarrow_{q,F,k}} = [x]_q [x-F_1]_q \cdots [x-F_{k-1}]_q$, $\overline{[x]}_{\downarrow_{q,F,k}}= [x]

Author

Bach, Quang T.; Paudyal, Roshil; Remmel, Jeffrey B.

Published

2017

Language

EN