Formula:KLS:09.10:02

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∑ x = 0 ∞ ( β ) x x ! ⁢ c x ⁢ M m ⁡ ( x ; β , c ) ⁢ M n ⁡ ( x ; β , c ) = c - n ⁢ n ! ( β ) n ⁢ ( 1 - c ) β ⁢ δ m , n superscript subscript 𝑥 0 Pochhammer-symbol 𝛽 𝑥 𝑥 superscript 𝑐 𝑥 Meixner-polynomial-M 𝑚 𝑥 𝛽 𝑐 Meixner-polynomial-M 𝑛 𝑥 𝛽 𝑐 superscript 𝑐 𝑛 𝑛 Pochhammer-symbol 𝛽 𝑛 superscript 1 𝑐 𝛽 Kronecker-delta 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\sum_{x=0}^{\infty}\frac{{\left(% \beta\right)_{x}}}{x!}c^{x}M_{m}\!\left(x;\beta,c\right)M_{n}\!\left(x;\beta,c% \right){}=\frac{c^{-n}n!}{{\left(\beta\right)_{n}}(1-c)^{\beta}}\,\delta_{m,n}% }}}

Constraint(s)

β > 0   0 < c < 1 formulae-sequence 𝛽 0 0 𝑐 1 {\displaystyle{\displaystyle{\displaystyle\beta>0\quad 0<c<1}}}


Proof

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Symbols List

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
M n subscript 𝑀 𝑛 {\displaystyle{\displaystyle{\displaystyle M_{n}}}}  : Meixner polynomial : http://dlmf.nist.gov/18.19#T1.t1.r9
δ m , n subscript 𝛿 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\delta_{m,n}}}}  : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4

Bibliography

Equation in Section 9.10 of KLS.

URL links

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