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Table of Integrals, Series, and Products, 8th Edition by Daniel Zwillinger

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9.33 Series of G-functions

Gpqmn(λx|b1,,bqa1,,ap)=λb1r=01r!(1λ)rGpqmn(x|b1+r,b2,,bqa1,,ap)[ | λ1 |<1, m1, if m=1 and p<q,λ may be arbitrary ]  =λbqr=01r!(λ1)rGpqmn(x|b1,,bq1,bq+ra1,,ap)[m<q, | λ1 |<1]=λa11r=01r!(λ1λ)rGpqmn(x|b1,bqa1r,a2,,ap)[ n1, Reλ<12,  (if n=1 and p>q, then λ may be arbitrary) ]=λap1r=01r!(1λ1)rGpqmn(x|b1,,bqa1,,ap1,apr)[ n<p, Reγ<12 ]

si331_e  EH I 213(1) EH I 213(2) EH I 213(3) EH I 213(4)

For integrals of the G-function, see 7.8.

9.34 Connections with other special functions

1. 

Jv(x)xμ=2μG0210(14x2|12v+12μ,12μ12v)

  EH I 219(44)

2. 

Yv(x)xμ=2μ=2μG1320(14x2|12μ12v,12μ+12v,12μ12v1212μ1

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