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I am trying to evaluate a fairly simple summation:

$\sum_{k=1}^n ka^kb^{n-k}$

Which is related to the common identity for $\sum_{k=1}^n ka^k$ available on Wikipedia.

I've previously seen lengthy lists of obscure summation formulas in the comments but could not find any this time via the search function.

I found Henry Goulds list at https://dokumen.tips/documents/combinatorial-identities-a-standardized-set-of-tables-listing-500-binomial.html, but it only has formulas involving binomial coefficients.

Is there a list of obscure summation identities? And in particular is there a published identity for this sum?

I got to this sum after some operations on a difference of powers factorization.

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  • $\begingroup$ Try out Wolfram Alpha. The code for your example is wolframalpha.com/input/… $\endgroup$
    – Johannes Trost
    Commented Aug 5, 2021 at 7:51
  • $\begingroup$ Thanks I hadn't thought of that! I did realize I could use the formula I listed by factoring out the $b^n$ term however. If you want to leave ur answer as an answer instead of a comment I will accept it, since it seems pretty general. $\endgroup$
    – summationman
    Commented Aug 5, 2021 at 7:55
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    $\begingroup$ $\sum_{k=1}^n ka^kb^{n-k}\ =\ b^n\cdot\sum_{k=1}^n\,k\cdot\left(\frac ab\right)^k$ $\endgroup$
    – Wlod AA
    Commented Aug 5, 2021 at 9:09
  • $\begingroup$ Thre are books of summation identities, e.g., Jolley, Summation of Series. books.google.com.au/books/about/… $\endgroup$
    – Gerry Myerson
    Commented Aug 5, 2021 at 10:02
  • $\begingroup$ See the freely available reference in this recent comment of mine. $\endgroup$
    – Dave L Renfro
    Commented Aug 5, 2021 at 15:02

1 Answer 1

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As suggested by the OP, I post my comment as an answer:

Try out Wolfram Alpha. The code for your example is https://www.wolframalpha.com/input/?i=Sum%5B+k+a+%5Ek+b%5E%28n-k%29+%2C+%7B+k%2C+1%2C+n+%7D+%5D

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