Formula:KLS:09.13:03

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∫ 0 ∞ x a ⁢ e - 2 x ⁢ y m ⁡ ( x ; a ) ⁢ y n ⁡ ( x ; a ) ⁢ 𝑑 x = - 2 a + 1 2 ⁢ n + a + 1 ⁢ Γ ⁡ ( - n - a ) ⁢ n ! ⁢ δ m , n superscript subscript 0 superscript 𝑥 𝑎 2 𝑥 Bessel-polynomial-y 𝑚 𝑥 𝑎 Bessel-polynomial-y 𝑛 𝑥 𝑎 differential-d 𝑥 superscript 2 𝑎 1 2 𝑛 𝑎 1 Euler-Gamma 𝑛 𝑎 𝑛 Kronecker-delta 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\int_{0}^{\infty}x^{a}{\mathrm{e}^{-% \frac{2}{x}}}y_{m}\!\left(x;a\right)y_{n}\!\left(x;a\right)\,dx=-\frac{2^{a+1}% }{2n+a+1}\Gamma\left(-n-a\right)n!\,\delta_{m,n}}}}

Constraint(s)

a < - 2 ⁢ N - 1 𝑎 2 𝑁 1 {\displaystyle{\displaystyle{\displaystyle a<-2N-1}}}


Proof

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Symbols List

∫ {\displaystyle{\displaystyle{\displaystyle\int}}}  : integral : http://dlmf.nist.gov/1.4#iv
e e {\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}  : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
y n subscript 𝑦 𝑛 {\displaystyle{\displaystyle{\displaystyle y_{n}}}}  : Bessel polynomial : http://dlmf.nist.gov/18.34#E1
Γ Γ {\displaystyle{\displaystyle{\displaystyle\Gamma}}}  : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
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 9.13 of KLS.

URL links

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