Formula:KLS:09.11:15

From DRMF
Jump to navigation Jump to search


[ ( 1 - t ) - γ ⁢ \HyperpFq ⁢ 21 ⁢ @ ⁢ @ ⁢ γ , - x - N ⁢ t p ⁢ ( t - 1 ) ] N = ∑ n = 0 N ( γ ) n n ! ⁢ K n ⁡ ( x ; p , N ) ⁢ t n subscript superscript 1 𝑡 𝛾 \HyperpFq 21 @ @ 𝛾 𝑥 𝑁 𝑡 𝑝 𝑡 1 𝑁 superscript subscript 𝑛 0 𝑁 Pochhammer-symbol 𝛾 𝑛 𝑛 Krawtchouk-polynomial-K 𝑛 𝑥 𝑝 𝑁 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\left[(1-t)^{-\gamma}\,\HyperpFq{2}{% 1}@@{\gamma,-x}{-N}{\frac{t}{p(t-1)}}\right]_{N}{}=\sum_{n=0}^{N}\frac{{\left(% \gamma\right)_{n}}}{n!}K_{n}\!\left(x;p,N\right)t^{n}}}}

Constraint(s)

γ 𝛾 {\displaystyle{\displaystyle{\displaystyle\gamma}}} arbitrary


Proof

We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.

Symbols List

F q p subscript subscript 𝐹 𝑞 𝑝 {\displaystyle{\displaystyle{\displaystyle{{}_{p}F_{q}}}}}  : generalized hypergeometric function : http://dlmf.nist.gov/16.2#E1
Σ Σ {\displaystyle{\displaystyle{\displaystyle\Sigma}}}  : sum : http://drmf.wmflabs.org/wiki/Definition:sum
( a ) n subscript 𝑎 𝑛 {\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}  : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii
K n subscript 𝐾 𝑛 {\displaystyle{\displaystyle{\displaystyle K_{n}}}}  : Krawtchouk polynomial : http://dlmf.nist.gov/18.19#T1.t1.r6

Bibliography

Equation in Section 9.11 of KLS.

URL links

We ask users to provide relevant URL links in this space.


Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Formula:KLS:09.11:15&oldid=3925"