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awebs4
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I'm trying to show that:
F(a, b; z) = F(a-1, b; z) + (z/b) F(a, b+1 ; z)
where F(a, b; z) is Kummer's confluent hypergeometric function and
F(a, b; z) = SUMn=0[ (a)n * z^n ] / [ (b)n * n!]
where (a)n is the Pochhammer symbol and is defined by:
a(a+1)(a+2)(a+3)...(a+n-1)
some Pochhammer identities include:
a(a+1)n = (a+n)(a)n = (a)n+1
my attempt includes pages of scrap that would be heinous to type out. i started on the right hand side of the equation and got it to look like:
[ (a-1)/(a+n-1) + z/(b+n) ] * F(a, b; z)
hoping that whatever was left after factoring out a F(a, b; z) would equal 1, but i can't get it to go away.
asking on here is really my last resort because I've spent so much time on this and it's due soon... so if someone's out there and can possibly help me, i hope you do it quickly. even if you help after it's due, i'd still like to know how it's done.
F(a, b; z) = F(a-1, b; z) + (z/b) F(a, b+1 ; z)
where F(a, b; z) is Kummer's confluent hypergeometric function and
F(a, b; z) = SUMn=0[ (a)n * z^n ] / [ (b)n * n!]
where (a)n is the Pochhammer symbol and is defined by:
a(a+1)(a+2)(a+3)...(a+n-1)
some Pochhammer identities include:
a(a+1)n = (a+n)(a)n = (a)n+1
The Attempt at a Solution
my attempt includes pages of scrap that would be heinous to type out. i started on the right hand side of the equation and got it to look like:
[ (a-1)/(a+n-1) + z/(b+n) ] * F(a, b; z)
hoping that whatever was left after factoring out a F(a, b; z) would equal 1, but i can't get it to go away.
asking on here is really my last resort because I've spent so much time on this and it's due soon... so if someone's out there and can possibly help me, i hope you do it quickly. even if you help after it's due, i'd still like to know how it's done.