Pochhammer symbols and Vandermonde's identity
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From: Pollux on 21 Jun 2010 17:47 Just wondering if there is a "nice" way to prove Vandermonde's identity by playing with Pochhammer symbols. Looked for identities involving Pochhammer symbols, but didn't managed to use those to prove Vandermonde's identity.
From: Francois Grondin on 22 Jun 2010 08:10 "Pollux" <po.lux(a)gmail.com> a ecrit dans le message de news: hvoml4$fpk$1(a)speranza.aioe.org... > Just wondering if there is a "nice" way to prove Vandermonde's identity > by playing with Pochhammer symbols. Looked for identities involving > Pochhammer symbols, but didn't managed to use those to prove > Vandermonde's identity. > The simplest way to demonstrate Vandermonde's identity is to start from (1-z)^(-a-b) = (1-z)^(-a) (1-z)^(-b). Express this last identity in series, knowing that : 1. (1-z)^(-a) = Hypergeometric1F0(a; z) = sum_k=0^infinity (Pochhammer(a, k) x^k / k!) 2. sum_k=0^infinity (sum_n=0^infinity f(k, n)) = sum_k=0^infinity (sum_n=0^k f(k, k-n)) . Compare the coefficients of z^n to get the desired result. Of course, you'll have to play with some properties of the Pochhammer symbol to reach the result. I don't know if there is another way to obtain it without comparing coefficients of z^n. Francois
From: Pollux on 22 Jun 2010 11:10 (6/22/10 5:10 AM), Francois Grondin wrote: > "Pollux"<po.lux(a)gmail.com> a ecrit dans le message de news: > hvoml4$fpk$1(a)speranza.aioe.org... >> Just wondering if there is a "nice" way to prove Vandermonde's identity >> by playing with Pochhammer symbols. Looked for identities involving >> Pochhammer symbols, but didn't managed to use those to prove >> Vandermonde's identity. >> > > The simplest way to demonstrate Vandermonde's identity is to start from > (1-z)^(-a-b) = (1-z)^(-a) (1-z)^(-b). Express this last identity in series, > knowing that : > > 1. (1-z)^(-a) = Hypergeometric1F0(a; z) = sum_k=0^infinity (Pochhammer(a, k) > x^k / k!) > 2. sum_k=0^infinity (sum_n=0^infinity f(k, n)) = sum_k=0^infinity (sum_n=0^k > f(k, k-n)) . > > Compare the coefficients of z^n to get the desired result. Of course, > you'll have to play with some properties of the Pochhammer symbol to reach > the result. > > I don't know if there is another way to obtain it without comparing > coefficients of z^n. > > Francois > > Thanks! Is there a reference where I can find identities for the Pochhammer symbols? I know they are related to the Gamma. Is it helpful to replace the Pochammer symbols with ratio of Gamma and work with the properties of the Gamma? Thanks again.
From: Francois Grondin on 22 Jun 2010 13:34 "Pollux" <po.lux(a)gmail.com> a ecrit dans le message de news: hvqjou$ptk$1(a)speranza.aioe.org... > (6/22/10 5:10 AM), Francois Grondin wrote: >> "Pollux"<po.lux(a)gmail.com> a ecrit dans le message de news: >> hvoml4$fpk$1(a)speranza.aioe.org... >>> Just wondering if there is a "nice" way to prove Vandermonde's identity >>> by playing with Pochhammer symbols. Looked for identities involving >>> Pochhammer symbols, but didn't managed to use those to prove >>> Vandermonde's identity. >>> >> >> The simplest way to demonstrate Vandermonde's identity is to start from >> (1-z)^(-a-b) = (1-z)^(-a) (1-z)^(-b). Express this last identity in >> series, >> knowing that : >> >> 1. (1-z)^(-a) = Hypergeometric1F0(a; z) = sum_k=0^infinity (Pochhammer(a, >> k) >> x^k / k!) >> 2. sum_k=0^infinity (sum_n=0^infinity f(k, n)) = sum_k=0^infinity >> (sum_n=0^k >> f(k, k-n)) . >> >> Compare the coefficients of z^n to get the desired result. Of course, >> you'll have to play with some properties of the Pochhammer symbol to >> reach >> the result. >> >> I don't know if there is another way to obtain it without comparing >> coefficients of z^n. >> >> Francois >> >> > Thanks! Is there a reference where I can find identities for the > Pochhammer symbols? I know they are related to the Gamma. Is it helpful > to replace the Pochammer symbols with ratio of Gamma and work with the > properties of the Gamma? > > Thanks again. Hi. You should have a look at Wolfram site (http://functions.wolfram.com/GammaBetaErf/Pochhammer/) as first reference. If you look for a book, I'd suggest "Special Functions" from E.D. Rainville, New York: Chelsea, 1971. You may also have a look at "Generalized Hypergeometric Functions" written by L.J. Slater (Cambridge, England: Cambridge University Press, 1966). In this last book, there is an extensive list of properties for the Pochhammer symbol. This should help. Regards. Francois
From: Pollux on 22 Jun 2010 13:39
Thanks a lot for the references. |