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Dilcher's Formula

sum_(1<=k<=n)(n; k)((-1)^(k-1))/(k^m)=sum_(1<=i_1<=i_2<=...<=i_m<=n)1/(i_1i_2...i_m),
(1)

where (n; k) is a binomial coefficient (Dilcher 1995, Flajolet and Sedgewick 1995, Prodinger 2000). An inverted version is given by

sum_(1<=k<=n)(n; k)(-1)^(k-1)sum_(1<=i_1<=i_2...<=i_m=k)1/(i_1i_2...i_m) =sum_(1<=k<=n)1/(k^m)=H_n^((m)),
(2)

where H_n^((k)) is a harmonic number of order m (Hernández 1999, Prodinger 2000). A q-analog of (1) is given by

sum_(1<=k<=n)[n; k]_q(-1)^(k-1)(q^((k+1; 2)+(m-1)k))/((1-q^k)^m) =sum_(1<=i_1<=i_2<=...<=i_m<=n)(q^(i_1))/(1-q^(i_1))...(q^(i_m))/(1-q^(i_m)),
(3)

where [n; k]_q is a q-binomial coefficient (Prodinger 2000).

See also

Binomial Identity

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References

Dilcher, K. "Some q-Series Identities Related to Divisor Functions." Disc. Math. 145, 83-93, 1995.Flajolet, P. and Sedgewick, R. "Mellin Transforms and Asymptotics: Finite Differences and Rice's Integrals." Theor. Comput. Sci. 144, 101-124, 1995.Hernández, V. "Solution IV of Problem 10490: A Reciprocal Summation Identity." Amer. Math. Monthly 106, 589-590, 1999.Prodinger, H. "A q-Analogue of a Formula of Hernandez Obtained by Inverting a Result of Dilcher." Austral. J. Combin. 21, 271-274, 2000.

Cite this as:

Weisstein, Eric W. "Dilcher's Formula." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DilchersFormula.html

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