Formula:KLS:09.05:21

From DRMF
Revision as of 08:34, 22 December 2019 by Move page script (talk | contribs) (Move page script moved page Formula:KLS:09.05:21 to F:KLS:09.05:21)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigation Jump to search


[ ( 1 - t ) - α - β - 1 ⁢ \HyperpFq ⁢ 32 ⁢ @ ⁢ @ ⁢ 1 2 ⁢ ( α + β + 1 ) , 1 2 ⁢ ( α + β + 2 ) , - x ⁢ α + 1 , - N - 4 ⁢ t ( 1 - t ) 2 ] N = ∑ n = 0 N ( α + β + 1 ) n n ! ⁢ Q n ⁡ ( x ; α , β , N ) ⁢ t n subscript superscript 1 𝑡 𝛼 𝛽 1 \HyperpFq 32 @ @ 1 2 𝛼 𝛽 1 1 2 𝛼 𝛽 2 𝑥 𝛼 1 𝑁 4 𝑡 superscript 1 𝑡 2 𝑁 superscript subscript 𝑛 0 𝑁 Pochhammer-symbol 𝛼 𝛽 1 𝑛 𝑛 Hahn-polynomial-Q 𝑛 𝑥 𝛼 𝛽 𝑁 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\left[(1-t)^{-\alpha-\beta-1}\,% \HyperpFq{3}{2}@@{\frac{1}{2}(\alpha+\beta+1),\frac{1}{2}(\alpha+\beta+2),-x}{% \alpha+1,-N}{-\frac{4t}{(1-t)^{2}}}\right]_{N}{}=\sum_{n=0}^{N}\frac{{\left(% \alpha+\beta+1\right)_{n}}}{n!}Q_{n}\!\left(x;\alpha,\beta,N\right)t^{n}}}}

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
Q n subscript 𝑄 𝑛 {\displaystyle{\displaystyle{\displaystyle Q_{n}}}}  : Hahn polynomial : http://dlmf.nist.gov/18.19#T1.t1.r3

Bibliography

Equation in Section 9.5 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:09.05:21&oldid=3507"