An extended general Hurwitz-Lerch zeta function as a Mathieu \((\mathbf a, \mathbf{\lambda})\)-series. (English) Zbl 1228.11135
Summary: It is shown that an integral representation for the extension of a general Hurwitz-Lerch zeta function recently obtained by M. Garg et al. [Appl. Appl. Math. 4, No. 1, 26–39, electronic only (2009; Zbl 1186.11055)] is a special case of the closed form integral expression for the Mathieu \((a, \lambda)\)-series given by T. K. Pogány [Integral Transforms Spec. Funct. 16, No. 8, 685–689 (2005; Zbl 1101.26018)]. As an immediate consequence of the derived results, new integral expressions and related bilateral bounding inequalities are investigated.
MSC:
| 11M35 | Hurwitz and Lerch zeta functions |
Keywords:
extended general Hurwitz-Lerch zeta function; Mathieu \((a, \lambda)\)-series; bilateral bounding inequalityReferences:
| [1] | Pogány, T. K., Integral representation of Mathieu \((a, \lambda)\)-series, Integral Transforms Spec. Funct., 16, no. 8, 685-689 (2005) · Zbl 1101.26018 |
| [2] | Feng, Qi, Inequalities for Mathieu series, RGMIA Research Report Collection, 4, No. 2, 187-193 (2001) |
| [3] | Cerone, P.; Lenard, C. T., On integral forms of generalized Mathieu series, JIPAM J. Inequal. Pure Appl. Math., 4, No. 5 (2003), art. 100 · Zbl 1072.26011 |
| [4] | Srivastava, H. M.; Tomovski, Ž., Some problems and solutions involving Mathieu’s series and its generalizations, JIPAM J. Inequal. Pure Appl. Math., 5, No. 2, 1-13 (2004), Art. 45 · Zbl 1068.33032 |
| [5] | Garg, M.; Jain, K.; Kalla, S. L., A further study of general Hurwitz-Lerch zeta function, Algebras Groups Geom., 25, no. 3, 311-319 (2008) · Zbl 1210.11096 |
| [6] | Garg, M.; Jain, K.; Kalla, S. L., On generalized Hurwitz-Lerch zeta distributions, Appl. Appl. Math., 4, No. 1, 26-39 (2009) · Zbl 1186.11055 |
| [7] | Pogány, T. K.; Srivastava, H. M.; Tomovski, Ž., Some families of Mathieu a-series and alternating Mathieu a-series, Appl. Math. Comput., 173, No. 1, 69-108 (2006) · Zbl 1097.33016 |
| [8] | (Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (1972), Dover: Dover New York), 9th printing · Zbl 0318.90001 |
| [9] | Elezović, N.; Giordano, C.; Pečarić, J., The best bounds in Gautschi’s inequality, Math. Inequal. Appl., 3, 239-252 (2000) · Zbl 0947.33001 |
| [10] | Feng Qi, Bai-Ni Guo, Sharp inequalities for the psi function and harmonic numbers, Available from: http://arxiv.org/abs/0902.2524; Feng Qi, Bai-Ni Guo, Sharp inequalities for the psi function and harmonic numbers, Available from: http://arxiv.org/abs/0902.2524 · Zbl 1294.33003 |
| [11] | Batir, N., Inequalities for the gamma function, Arch. Math., 91, 554-563 (2008) · Zbl 1165.33001 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
