Product Formulas of Watson, Bailey and Bateman Types and Positivity of the Poisson Kernel for q-Racah Polynomials

1984 ◽  
Vol 15 (4) ◽  
pp. 768-789 ◽  
Author(s):  
George Gasper ◽  
Mizan Rahman
Keyword(s):  
Poisson Kernel ◽  
Racah Polynomials ◽  
Product Formulas

scholarly journals Nonnegative kernels in product formulas for q-Racah polynomials. I

1983 ◽  
Vol 95 (2) ◽  
pp. 304-318 ◽  
Author(s):  
George Gasper ◽  
Mizan Rahman
Keyword(s):  
Racah Polynomials ◽  
Product Formulas

Harmonic functions; Poisson kernel

2013 ◽  
pp. 28-51
Author(s):  
Camil Muscalu ◽  
Wilhelm Schlag
Keyword(s):  
Harmonic Functions ◽  
Poisson Kernel

scholarly journals Multi-indexed (q-)Racah polynomials

2012 ◽  
Vol 45 (38) ◽  
pp. 385201 ◽  
Author(s):  
Satoru Odake ◽  
Ryu Sasaki
Keyword(s):  
Racah Polynomials

The Bessel–Poisson Kernel

2013 ◽  
pp. 95-100
Keyword(s):  
Poisson Kernel

scholarly journals Dual Addition Formulas Associated with Dual Product Formulas

2018 ◽  
pp. 373-392
Author(s):  
Tom H. Koornwinder
Keyword(s):  
Addition Formulas ◽  
Product Formulas

scholarly journals Poisson kernel for the associated continuous $q$-ultraspherical polynomials

1997 ◽  
Vol 4 (1) ◽  
pp. 77-90 ◽  
Author(s):  
Mizan Rahman ◽  
Qazi M. Tariq
Keyword(s):  
Poisson Kernel ◽  
Ultraspherical Polynomials

scholarly journals Augmented Rook Boards and General Product Formulas

10.37236/809 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Brian K. Miceli ◽  
Jeffrey Remmel
Keyword(s):  
Product Formula ◽  
Combinatorial Proof ◽  
Rook Theory ◽  
Special Cases ◽  
Ferrers Board ◽  
General Product ◽  
Product Formulas

There are a number of so-called factorization theorems for rook polynomials that have appeared in the literature. For example, Goldman, Joichi and White showed that for any Ferrers board $B = F(b_1, b_2, \ldots, b_n)$, $$\prod_{i=1}^n (x+b_i-(i-1)) = \sum_{k=0}^n r_k(B) (x)\downarrow_{n-k}$$ where $r_k(B)$ is the $k$-th rook number of $B$ and $(x)\downarrow_k = x(x-1) \cdots (x-(k-1))$ is the usual falling factorial polynomial. Similar formulas where $r_k(B)$ is replaced by some appropriate generalization of the $k$-th rook number and $(x)\downarrow_k$ is replaced by polynomials like $(x)\uparrow_{k,j} = x(x+j) \cdots (x+j(k-1))$ or $(x)\downarrow_{k,j} = x(x-j) \cdots (x-j(k-1))$ can be found in the work of Goldman and Haglund, Remmel and Wachs, Haglund and Remmel, and Briggs and Remmel. We shall refer to such formulas as product formulas. The main goal of this paper is to develop a new rook theory setting in which we can give a uniform combinatorial proof of a general product formula that includes, as special cases, essentially all the product formulas referred to above. We shall also prove $q$-analogues and $(p,q)$-analogues of our general product formula.

scholarly journals Lozenge Tiling Function Ratios for Hexagons with Dents on Two Sides

10.37236/9363 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Daniel Condon
Keyword(s):  
Triangular Lattice ◽  
Lozenge Tilings ◽  
Simple Product ◽  
Two Sides ◽  
Product Formulas

We give a formula for the number of lozenge tilings of a hexagon on the triangular lattice with unit triangles removed from arbitrary positions along two non-adjacent, non-opposite sides. Our formula implies that for certain families of such regions, the ratios of their numbers of tilings are given by simple product formulas.

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