Formula:KLS:09.12:12

From DRMF
Jump to navigation Jump to search


e t ⁢ \HyperpFq ⁢ 01 ⁢ @ ⁢ @ - α + 1 - x ⁢ t = ∑ n = 0 ∞ L n α ⁡ ( x ) ( α + 1 ) n ⁢ t n 𝑡 \HyperpFq 01 @ @ 𝛼 1 𝑥 𝑡 superscript subscript 𝑛 0 generalized-Laguerre-polynomial-L 𝛼 𝑛 𝑥 Pochhammer-symbol 𝛼 1 𝑛 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle{\mathrm{e}^{t}}\,\HyperpFq{0}{1}@@{% -}{\alpha+1}{-xt}=\sum_{n=0}^{\infty}\frac{L^{\alpha}_{n}\left(x\right)}{{% \left(\alpha+1\right)_{n}}}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

e e {\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}  : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
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
L n ( α ) superscript subscript 𝐿 𝑛 𝛼 {\displaystyle{\displaystyle{\displaystyle L_{n}^{(\alpha)}}}}  : Laguerre (or generalized Laguerre) polynomial : http://dlmf.nist.gov/18.3#T1.t1.r27
( a ) n subscript 𝑎 𝑛 {\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}  : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii

Bibliography

Equation in Section 9.12 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:09.12:12&oldid=3993"