A Summation from A Generalized Negative Binomial Distribution












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I am reading Jain and Consul "1971A Generalized Negative Binomial Distribution". The key identity of this generalised negative binomial distribution is (slightly different version):



$$(1-alpha )^{-n}=sum _{x=0}^{infty } frac{n}{n+x beta }frac{Gamma(n+x beta +1)}{Gamma(x+1)Gamma(n+x (beta
-1)+1)}alpha ^x(1-alpha )^{x(beta -1)}, quad 0<alpha<1.$$



My question is that, apart from Lagrange's formula(see paper), is there any other way to show this identity?



Thanks in advance.






summation taylor-expansion negative-binomial







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$endgroup$

















    2












    $begingroup$


    I am reading Jain and Consul "1971A Generalized Negative Binomial Distribution". The key identity of this generalised negative binomial distribution is (slightly different version):



    $$(1-alpha )^{-n}=sum _{x=0}^{infty } frac{n}{n+x beta }frac{Gamma(n+x beta +1)}{Gamma(x+1)Gamma(n+x (beta
    -1)+1)}alpha ^x(1-alpha )^{x(beta -1)}, quad 0<alpha<1.$$



    My question is that, apart from Lagrange's formula(see paper), is there any other way to show this identity?



    Thanks in advance.






    summation taylor-expansion negative-binomial







    share|cite|improve this question











    $endgroup$















      2












      2








      2





      $begingroup$


      I am reading Jain and Consul "1971A Generalized Negative Binomial Distribution". The key identity of this generalised negative binomial distribution is (slightly different version):



      $$(1-alpha )^{-n}=sum _{x=0}^{infty } frac{n}{n+x beta }frac{Gamma(n+x beta +1)}{Gamma(x+1)Gamma(n+x (beta
      -1)+1)}alpha ^x(1-alpha )^{x(beta -1)}, quad 0<alpha<1.$$



      My question is that, apart from Lagrange's formula(see paper), is there any other way to show this identity?



      Thanks in advance.






      summation taylor-expansion negative-binomial







      share|cite|improve this question











      $endgroup$




      I am reading Jain and Consul "1971A Generalized Negative Binomial Distribution". The key identity of this generalised negative binomial distribution is (slightly different version):



      $$(1-alpha )^{-n}=sum _{x=0}^{infty } frac{n}{n+x beta }frac{Gamma(n+x beta +1)}{Gamma(x+1)Gamma(n+x (beta
      -1)+1)}alpha ^x(1-alpha )^{x(beta -1)}, quad 0<alpha<1.$$



      My question is that, apart from Lagrange's formula(see paper), is there any other way to show this identity?



      Thanks in advance.






      summation taylor-expansion negative-binomial




      summation taylor-expansion negative-binomial






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Jan 31 at 20:50









      Renato Faraone

      2,36611727




      2,36611727










      asked Jan 26 at 5:41









      gouwangzhangdonggouwangzhangdong

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      888






















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