[math-fun] Harmonic numbers
H(n) = SUM(1/j, j=1..n) is the nth "harmonic number." QUESTION: what if we want to generalize this to real or complex n, not merely integer n, similarly to how Euler converted factorial to gamma function? ANSWER: Let psi(x) = d/dx ( ln Gamma(x) ) = Gamma'x) / Gamma(x) be the "psi function"; then H(x) = psi(x) + EulerMascheroniConstant = 0.5772156649 is the sought-for generalization. Euler also had found the integral representation H(x) = INTEGRAL( (1 - t^x) / (1-t), t=0..1 ) which is equivalent to this. http://en.wikipedia.org/wiki/Harmonic_number#Special_values_for_fractional_a... discusses further, and many other things are "continuizable" in similar way (also discussed in the wikipedia article).
The summer after high school I found the formula: Sum_{k=1..n} 1/k^s = (1/Gamma(s))*Integral_{0..1} ((1-x^n)/(1-x))* ln(1/x)^(s-1) dx [if I’m remembering correctly], which can be extended on the RHS to at least any real s > 1 and probably lots more complex s as well. (As I’ve mentioned, I showed this to my freshman advisor, who reached over and pointed out the same formula in his Whittaker & Watson.) —Dan On Mar 3, 2014, at 2:42 PM, Warren D Smith <warren.wds@gmail.com> wrote:
H(n) = SUM(1/j, j=1..n) is the nth "harmonic number."
QUESTION: what if we want to generalize this to real or complex n, not merely integer n, similarly to how Euler converted factorial to gamma function? . . . Euler also had found the integral representation
H(x) = INTEGRAL( (1 - t^x) / (1-t), t=0..1 )
which is equivalent to this.
On Mon, Mar 3, 2014 at 3:42 PM, Warren D Smith <warren.wds@gmail.com> wrote:
Euler also had found the integral representation H(x) = INTEGRAL( (1 - t^x) / (1-t), t=0..1 ) which is equivalent to this.
The integrand is the q-deformation of x where q = t. Is there any combinatorical content there? -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
participants (3)
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Dan Asimov -
Mike Stay -
Warren D Smith