# Generalized harmonic number function

I saw the following definition in a paper

$$\tag{1} H^{(m)}_n(x)=\sum^{n-1}_{k=0}\frac{1}{(k+x)^m}$$

Is this widely used ? or only by choice of the author ?

See page number 3 of the paper

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This can also be interpreted as a truncated Hurwitz Zeta function: en.m.wikipedia.org/wiki/Hurwitz_zeta_function –  Alex R. Aug 6 '13 at 17:14
@Alex I know of that , but my interest is whether this is widely used as a symbol because I used this symbol for another purpose ! –  Zaid Alyafeai Aug 6 '13 at 17:19
Well Knuth used much $\,\displaystyle H^{(m)}_n=\sum^{n}_{k=1}\frac{1}{k^m}\,$ in TAOCP as well as in 'Concrete Mathematics' to generalize the harmonic sum so that this choice seems rather coherent. But there is at least a conflict with the Hankel function as seen in G&R, Wikipedia and others. –  Raymond Manzoni Aug 6 '13 at 18:16
@Raymond Hankel functions seem to only use the notations $H^{(1)}_n(x),H^{(2)}_n(x)$ , right ? –  Zaid Alyafeai Aug 6 '13 at 18:24
Yes but you don't exclude $1$ and $2$ for $m$... –  Raymond Manzoni Aug 6 '13 at 18:27