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i noticed that there are two formulas for the series representation of $(1+x)^n$, which are:

1) $\sum_{k=0}^\infty {{n}\choose{k}} x^k$

2) $\sum_{k=0}^\infty \frac{(-n)_k}{k!} (-x)^k$

What is the difference between them? which one is the correct one? all i want is to decide which formula to use only, i have ${n}\choose k$, where n equals $m/2+d$, where m is a positive number, and d is the index of a summation from 0 to $\infty$

user42138
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  • What does $(-n)_{k}$ represent? – S.C.B. Feb 05 '17 at 11:40
  • @S.C.B. it's the Pochhammer symbol – user42138 Feb 05 '17 at 11:41
  • ${n\choose k}=\frac{n(n-1)\cdots(n-k+1)}{k!}$ and $(-n)_k=(-n)(-n+1)\cdots(-n+k-1)$; so there is no difference at all ! – Adren Feb 05 '17 at 11:53
  • @Adren
    in the first definition, in the (nk)(nk), i get negative value inside the factorial(n-k), that makes a problem, but if i use the second difinition, i replace (−n)k(−n)k by (−1)k(n−k+1)k(−1)k(n−k+1)k and it works, so there must be a difference, right?     – user42138 Feb 05 '17 at 12:00
  • @user42138: you should consider that ${b\choose a}$ denotes here the so-called generalized binomial coefficient. So the relative values of $a$ and $b$ are not causing any problem. Also, $a$ doesn't have to be integer. It doesn't happen here, though, but does happen in $\sum_{k=0}^\infty{1/2\choose k}x^k$ – Adren Feb 05 '17 at 12:18
  • A probabilistic connection, more precisely with probability generating functions (pgf)s (https://www.stat.auckland.ac.nz/~fewster/325/notes/ch4.pdf), wishing you have had lectures on the subject; with an approximate normalization by $1/2^n$, the first one is the pgf of a binomial Bin(n,1/2), the second one is the pgf of a Pascal(n,1/2) (http://math.stackexchange.com/q/1071163) – Jean Marie Feb 05 '17 at 12:31

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