Formula:KLS:09.10:11

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∇ ⁡ [ ( β ) x ⁢ c x x ! ⁢ M n ⁡ ( x ; β , c ) ] = ( β - 1 ) x ⁢ c x x ! ⁢ M n + 1 ⁡ ( x ; β - 1 , c ) ∇ Pochhammer-symbol 𝛽 𝑥 superscript 𝑐 𝑥 𝑥 Meixner-polynomial-M 𝑛 𝑥 𝛽 𝑐 Pochhammer-symbol 𝛽 1 𝑥 superscript 𝑐 𝑥 𝑥 Meixner-polynomial-M 𝑛 1 𝑥 𝛽 1 𝑐 {\displaystyle{\displaystyle{\displaystyle\nabla\left[\frac{{\left(\beta\right% )_{x}}c^{x}}{x!}M_{n}\!\left(x;\beta,c\right)\right]=\frac{{\left(\beta-1% \right)_{x}}c^{x}}{x!}M_{n+1}\!\left(x;\beta-1,c\right)}}}

Proof

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Symbols List

( a ) n subscript 𝑎 𝑛 {\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}  : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii
M n subscript 𝑀 𝑛 {\displaystyle{\displaystyle{\displaystyle M_{n}}}}  : Meixner polynomial : http://dlmf.nist.gov/18.19#T1.t1.r9

Bibliography

Equation in Section 9.10 of KLS.

URL links

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