The Lerch zeta and related functions of non-positive integer order

Cvijović, Djurdje

doi:10.1090/S0002-9939-09-10116-8

The Lerch zeta and related functions of non-positive integer order
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Proc. Amer. Math. Soc. 138 (2010), 827-836 Request permission

Abstract:

It is known that the Lerch (or periodic) zeta function of non-positive integer order, $\ell _{-n}(\xi )$, $n\in \mathbb {N}_{0}:= \{0, 1, 2, 3, \ldots \}$, is a polynomial in $\cot (\pi \xi )$ of degree $n + 1$. In this paper, a very simple explicit closed-form formula for this polynomial valid for any degree is derived. In addition, novel analogous explicit closed-form formulae for the Legendre chi function, the alternating Lerch zeta function and the alternating Legendre chi function are established. The obtained formulae involve the Carlitz-Scoville tangent and secant numbers of higher order, and the derivative polynomials for tangent and secant are used in their derivation. Several special cases and consequences are pointed out, and some examples are also given.

References

T. M. Apostol, On the Lerch zeta function, Pacific J. Math. 1 (1951), 161–167. MR 43843, DOI 10.2140/pjm.1951.1.161
Tom M. Apostol, Dirichlet $L$-functions and character power sums, J. Number Theory 2 (1970), 223–234. MR 258766, DOI 10.1016/0022-314X(70)90022-3
Tom M. Apostol, Introduction to analytic number theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1976. MR 0434929
Khristo N. Boyadzhiev, Apostol-Bernoulli functions, derivative polynomials and Eulerian polynomials, Adv. Appl. Discrete Math. 1 (2008), no. 2, 109–122. MR 2450260
L. Carlitz and Richard Scoville, Tangent numbers and operators, Duke Math. J. 39 (1972), 413–429. MR 302825, DOI 10.1215/S0012-7094-72-03949-X
L. Carlitz, Permutations, sequences and special functions, SIAM Rev. 17 (1975), 298–322. MR 371673, DOI 10.1137/1017034
C.-H. Chang and C.-W. Ha, Central factorial numbers and values of Bernoulli and Euler polynomials at rationals, Num. Func. Anal. Optim. 30 (2009), 214–226.
Djurdje Cvijović and Jacek Klinowski, Values of the Legendre chi and Hurwitz zeta functions at rational arguments, Math. Comp. 68 (1999), no. 228, 1623–1630. MR 1648375, DOI 10.1090/S0025-5718-99-01091-1
Djurdje Cvijović, Integral representations of the Legendre chi function, J. Math. Anal. Appl. 332 (2007), no. 2, 1056–1062. MR 2324319, DOI 10.1016/j.jmaa.2006.10.083
Michael E. Hoffman, Derivative polynomials for tangent and secant, Amer. Math. Monthly 102 (1995), no. 1, 23–30. MR 1321452, DOI 10.2307/2974853
Michael E. Hoffman, Derivative polynomials, Euler polynomials, and associated integer sequences, Electron. J. Combin. 6 (1999), Research Paper 21, 13. MR 1685701
Donald E. Knuth and Thomas J. Buckholtz, Computation of tangent, Euler, and Bernoulli numbers, Math. Comp. 21 (1967), 663–688. MR 221735, DOI 10.1090/S0025-5718-1967-0221735-9
M. Lerch, Note sur la fonction ${\mathfrak {K}} \left ( {w,x,s} \right ) = \sum \limits _{k = 0}^\infty {\frac {{e^{2k\pi ix} }}{{\left ( {w + k} \right )^s }}}$, Acta Math. 11 (1887), no. 1-4, 19–24 (French). MR 1554747, DOI 10.1007/BF02418041
Earl D. Rainville, Special functions, The Macmillan Company, New York, 1960. MR 0107725
H. M. Srivastava and Junesang Choi, Series associated with the zeta and related functions, Kluwer Academic Publishers, Dordrecht, 2001. MR 1849375, DOI 10.1007/978-94-015-9672-5