Extension of Mathieu series and alternating Mathieu series involving the Neumann function Yν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_\nu $$\end{document}

The main objective of this paper is to present a new extension of the familiar Mathieu series and the alternating Mathieu series S(r) and S~(r)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\widetilde{S}}}(r)$$\end{document} which are denoted by Sμ,ν(r)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {S}}_{\mu ,\nu }(r)$$\end{document} and S~μ,ν(r)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde{{\mathbb {S}}}_{\mu ,\nu }(r)$$\end{document}, respectively. The computable series expansions of their related integral representations are obtained in terms of the exponential integral E1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_1$$\end{document}, and convergence rate discussion is provided for the associated series expansions. Further, for the series Sμ,ν(r)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {S}}_{\mu ,\nu }(r)$$\end{document} and S~μ,ν(r)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde{{\mathbb {S}}}_{\mu ,\nu }(r)$$\end{document}, related expansions are presented in terms of the Riemann Zeta function and the Dirichlet Eta function, also their series built in Gauss’ 2F1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}_2F_1$$\end{document} functions and the associated Legendre function of the second kind Qμν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_\mu ^\nu $$\end{document} are given. Our discussion also includes the extended versions of the complete Butzer–Flocke–Hauss Omega functions. Finally, functional bounding inequalities are derived for the investigated extensions of Mathieu-type series.


Introduction and preliminaries
During the study of elasticity of solid bodies, Émile Leonard Mathieu (1835-1890) introduced and investigated the famous infinite functional series called Mathieu series of the form [20] S(r ) = n≥1 2n (n 2 + r 2 ) 2 , r > 0.
The alternating version of Mathieu series, introduced and investigated by Pogány et al. in [27, p. 72, Eq. (2.7)], is Elegant integral forms of the Mathieu series S(r ) and the alternating Mathieu series S(r ) was established by Emersleben [13]: Milovanović and Pogány [22] discovered other integral forms for Mathieu and alternating Mathieu series; Tomovski and Pogány [29] deduced Cauchy principal value integrals for these series; moreover, see [7,9,12] for this integral form, and [8,25,26] for another similarly focused study. The present authors studied and investigated a multi-parameter extension of the well-known Mathieu series and the alternating Mathieu series in a recent paper [24]. We emphasize the integral representations [22,  which will have a special treatment below. Let N, Z, and C be the sets of positive integers, integers, and complex numbers, respectively. The Bessel function of the first kind of the order ν is defined by where the principal branch of J ν (z) should be considered (it corresponds to the principal value of z ν ) and J ν (z) is analytic in the z-plane cut along the interval (−∞, 0]. Moreover, for ν ∈ Z, the Bessel function of the first kind is entire in z on the whole complex plane, see [15, p. 5]. The Bessel function of the second kind (Neumann function or Weber-Bessel function) of order ν is expressible in terms of the Bessel function of the first kind defined as [30, p. 64)] On the other hand, see [23, p. 228, Eq. (10.16.1)], We can realize the extension of the Mathieu series by considering the related integral representation extending the integrand by a weight function. Namely, rewrite (1.1) into the form The same can be done for the alternating Mathieu series, so

Polylogarithmic approach to Mathieu and alternating Mathieu series
In the exposition we use the series definition of the Riemann Zeta function [28, p. 164 The close relative of the Riemann Zeta function known as the Dirichlet Eta function (or the alternating Riemann Zeta function) η(s) and its integral representation are given by [28,p. 384,Eq. (35)] respectively. The polylogarithm (de Jonquière's function) is the Dirichlet type power series in complex argument z, viz.
Li s (z) = n≥1 z n n s ; here the defining series converges for the complex order s ∈ C for all |z| < 1, while by analytic continuation it can be extended to |z| ≥ 1. There is extensive literature available for the polylogarithm and related topic; consult the standard references [1,14,19,23,31]. Obviously, Li s (1) = ζ(s), (s) > 1.
Our interest in polylogarithm is drawn by the integral representation This integral is closely connected with the Bose-Einstein distribution's integral [10] Here x ≤ 0; in turn for x > 0 the Cauchy principal value integral should be used, see [10]. Obviously, (2.4) The Fermi-Dirac distribution integral (see also Clunie's note [10]) is We point out, see [31], that The similarity to Emersleben's integral expressions for the Mathieu series and the alternating Mathieu series S(r ) is obvious, compare (1.1) and (1.2). Motivated by these 'similarities', our next goal is to establish inter-connection formulae between the polylogarithm, the series built from the Riemann Zeta function, the Fermi-Dirac and Bose-Einstein integrals from one, and Mathieu series and alternating Mathieu series from the other side.

Theorem 2.1
For all |r | < 1, Proof Consider the integral representation (1.1). By the Taylor expansion of the sine function in the integrand we conclude In turn, by (2.3) and (2.4) we confirm that

8) which results in
getting (2.6). Next, starting now from (1.2) we infer similarly the second formula which holds for the alternating Mathieu series S(r ). Indeed, applying (2.5), we conclude which completes the proof.

Series expansions of integrals (1.3) and (1.4)
The derivation of the integral expressions (1.3) and (1.4) associated to S(r ) and S(r ) is realized by complex analytical and integral transformation methods, see [22]. Then, since their integrands include reciprocals of hyperbolic functions, we explore other series expansions of these integrals. First, we introduce the exponential integral of the first order [1, p. 228, Eq. 5.1.1] whose mirror symmetry property reads

2)
where [z] denotes the real part of z ∈ C.
Proof Expanding the secant hyperbolic kernel in the integrand of (1.3), for all x > 0 we have Next, we need the related Laplace integral property [1, p. 230, Eq. 5.
The partial fraction decomposition of the integrand is Hence, applying the previously listed results, we have By the mirror symmetry of the exponential integral we readily conclude whose right-hand side for s = 2π(n + 1) reduces to Inserting the last expression into (3.4) we arrive at the series expansion (3.1). Next, as to (3.2), since the integral expression (1.4) becomes the following series of Laplace transforms: The partial fraction decomposition of the Laplace transform input function reads Again, by the mirror symmetry of the exponential integral E 1 (z), inserting s = π(2n + 1), we conclude that The rest is obvious.
Unfortunately, the series (3.1) for the sum S(r ) is slowly convergent. Denote its general term by u n (r ), i.e., , and consider another auxiliary series: where n ≥ 0 and r > 0. Let S n (r ) and T n (r ) be nth partial sums of the series S(r ) and T (r ), respectively. Since the series (3.1) is convergent, lim n→+∞ u n (r ) = 0, and according to T n (r ) − S n (r ) = 1 2 u n+1 (r ) we conclude that T (r ) is also a convergent series with the same sum S(r ).

Remark 3.2
Numerical calculations show that for fixed values of r , u n (r ) > 0 for even n, and negative for odd n, so the transformation of the series (3.1) given by (3.5) is, in fact, the well-known Euler-Abel transformation. The series (3.1) is extremely slowly convergent and is practically not usable for numerical calculations. On the other hand, the transformed series (3.5) shows a relatively fast convergence, so a reasonable number of initial terms is enough to approximate the sum S(r ) with the required accuracy. The following examples illustrate these properties. Fig. 1 (left) we present the errors

Example 3.3 In
with only n = 0, 1, 2, and 5. As the exact value S(r ) we take a very precise approximation obtained by using the Gaussian quadrature formula with respect to the hyperbolic weight function (see [21,22]), applied directly to the integral (1.3). As we can see, only for small values of r , the errors E S,n (r ) are significant if n ≤ 5. In the same figure (right) we present the corresponding relative errors R S,n (r ) = |E S,n (r )/S(r )|, taking the partial sums in (3.6) for n = 5, 10, 50 and 100 terms. For example, with n = 100, the relative error for r ∈ [0, 1] is less than 10 −6 , and for larger r > 1 this error is less than 10 −8 , which means that we obtain the values of S(r ) with at least 6 and 8 exact decimal digits, respectively. A series with faster convergence can be obtained by repeating the previous transformation to the series (3.5). Then we get The corresponding errors in the partial sums are denoted by E S,n (r ) and presented in Fig. 2 (left), as well as the relative errors R S,n (r ) in the same figure (right). .

Example 3.4 In the case of the alternating Mathieu series S(r ) we study the auxiliary series
The repeated Euler-Abel transformation, in this case, gives the accelerated series in the following form: The corresponding diagrams are presented in Figs. 3 and 4 with the same notations as the ones in the previous case for the sum S(r ) (Example 3.3).

Remark 3.5
As we can see, there exist certain oscillations in the graphics for the relative errors R S,n (r ) (Fig. 2 (right)) and RS ,n (r ) (Fig. 4 (right)) for larger r and sufficiently large n (n = 100), because of unstable calculations in such cases. Namely, the values S(r ) andS(r ), as well as their approximations, i.e., the partial sums of series (3.7) and (3.9), respectively, are close to zero in such cases.

The extended Mathieu series S , (r) and S , (r)
Motivated by (1.7), replacing there the kernel function Y − 1 2 with the general Bessel function of the second kind of order ν, we introduce the extended Mathieu series S(r ) and its alternating variant S(r ) in the following forms: where in both cases r > 0, μ > 0. Clearly we obtain the following recurrence formulae: Theorem 4.1 If μ, ν + 1 > 0, n ∈ N and μ > |ν| > 0, then Moreover, if |ν| < 1 and μ + ν + 1 > 0, then .
Proof Insert the binomial series expansion (e x − 1) −1 = n≥1 e −nx , x > 0 into (4.1). The legitimate integral-sum interchange which can be proved, e.g., by the dominated convergence theorem results in whose parameter space consists of (μ) > | (ν)| and (a ± ib) > 0, taken above a = n and b = r , we conclude the first asserted formula.
In the sequel we need the associated Legendre function of the second kind of a real argument [23, Eq. 14.3.7] provided the parameter range consists of p, q ∈ C and −( p + q) / ∈ N.
Theorem 4.2 If μ, ν + 1 > 0, n ∈ N, and μ > |ν| > 0, then Proof The same binomial expansion as in the previous proof and a change of the order of integration and summation gives By virtue of the integral [17, p. 700, Eq. 6.621 whose parameter space consists of a > 0, b > 0, (μ) > | (ν)|, for a = n and b = r we obtain the first asserted formula. The derivation of the series expansion for S μ,ν (r ) gives Now, the path to the final formula is obvious.

Functional bounding inequalities
Recall the Gubler-Weber formula [30,p. 165,Eq. (5)] which holds for (z) > 0 and ν > −1/2. Splitting the ν-domain into three disjoint intervals Baricz et al. [3, pp. 957-958] obtained the functional bounding inequality for the real argument Neumann function Y ν (x) (see also [18, pp. 7-8], [11, p. 76]): (5.1) Moreover, if μ + ν + 1 > 0, then , , Proof Starting with (4.1) and splitting the range of ν into three disjoint intervals and using the estimates (5.1), we conclude which is equivalent to the first statement of this theorem. In the derivation procedure we apply the integral representation (2.1) of the Riemann Zeta function. Similarly, if we start with the expression (4.2), we obtain the second formula with the aid of the Dirichlet Eta function's integral form (2.2). In both cases the parameter constraints are controlled by the convergence conditions (2.1) and (2.2), respectively.
For a noninteger order ν / ∈ Z there exist several equivalent series representations; we work with the reformulated (1.6), viz.
Proof Applying the Mellin transform for all integrals which we derive by the lines of the previous proof, we clearly deduce the claimed result. Now, we present the Riemann Zeta building blocks series presentation of the extended Mathieu S μ,ν (r ) and Dirichlet Eta function terms for extended alternating Mathieu series S μ,ν (r ) by using the noninteger ν parameter case.

Ω(x) = 2
Funding Open access funding provided by óbuda University.
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