Abstract
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 \({{\widetilde{S}}}(r)\) which are denoted by \({\mathbb {S}}_{\mu ,\nu }(r)\) and \(\widetilde{{\mathbb {S}}}_{\mu ,\nu }(r)\), respectively. The computable series expansions of their related integral representations are obtained in terms of the exponential integral \(E_1\), and convergence rate discussion is provided for the associated series expansions. Further, for the series \({\mathbb {S}}_{\mu ,\nu }(r)\) and \(\widetilde{{\mathbb {S}}}_{\mu ,\nu }(r)\), related expansions are presented in terms of the Riemann Zeta function and the Dirichlet Eta function, also their series built in Gauss’ \({}_2F_1\) functions and the associated Legendre function of the second kind \(Q_\mu ^\nu \) 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.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 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]
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 \({\widetilde{S}}(r)\) was established by Emersleben [13]:
and Pogány et al. [27, p. 72, Eq. (2.8)]:
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, pp. 185–186, Corollary 2.2]
which will have a special treatment below.
Let \({\mathbb {N}}, {\mathbb {Z}}\), and \({\mathbb {C}}\) be the sets of positive integers, integers, and complex numbers, respectively. The Bessel function of the first kind of the order \(\nu \) is defined by
where the principal branch of \(J_\nu (z)\) should be considered (it corresponds to the principal value of \(z^\nu \)) and \(J_\nu (z)\) is analytic in the z-plane cut along the interval \((-\infty , 0]\). Moreover, for \(\nu \in {\mathbb {Z}}\), 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 \(\nu \) is expressible in terms of the Bessel function of the first kind defined as [30, p. 64)]
Also, Bessel functions of half-integer order have the connection or recurrence formula [17, p. 925, Eq. (8.465)]
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
2 Polylogarithmic approach to Mathieu and alternating Mathieu series
In the exposition we use the series definition of the Riemann Zeta function [28, p. 164, Eq. (1)]
and its integral representation
The close relative of the Riemann Zeta function known as the Dirichlet Eta function (or the alternating Riemann Zeta function) \(\eta (s)\) and its integral representation are given by [28, p. 384, Eq. (35)]
that is,
respectively.
The polylogarithm (de Jonquière’s function) is the Dirichlet type power series in complex argument z, viz.
here the defining series converges for the complex order \(s \in {\mathbb {C}}\) for all \(|z|<1\), while by analytic continuation it can be extended to \(|z| \ge 1\). There is extensive literature available for the polylogarithm and related topic; consult the standard references [1, 14, 19, 23, 31]. Obviously, \({\mathrm{Li}}_s(1) = \zeta (s),\,\Re (s)>1\).
Our interest in polylogarithm is drawn by the integral representation
This integral is closely connected with the Bose–Einstein distribution’s integral [10]
Here \(x \le 0\); in turn for \(x>0\) the Cauchy principal value integral should be used, see [10]. Obviously,
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 \({{\widetilde{S}}}(r)\) 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
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 \({{\widetilde{S}}}(r)\). Indeed, applying (2.5), we conclude
which completes the proof. \(\square \)
Remark 2.2
We point out that (2.6) and (2.7) are not new; in fact these relations coincide with the series representations [24, Eqs. (1.7-8)], also see [27, p. 72, Proposition 1] for (2.7). We also point out that there are no reasons to consider the series S(r) and \({{\widetilde{S}}}(r)\) exclusively for \(r>0\); the exception can be Mathieu’s original mathematical model in which he described the vibration of clamped rectangular plates and membranes, see the discussion in the memoir [24, §8.3]. Hence the importance of the previously presented results.
3 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 \({{\widetilde{S}}}(r)\) 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]
![](http://media.springernature.com/lw275/springer-static/image/art%3A10.1007%2Fs10998-022-00471-9/MediaObjects/10998_2022_471_Equ94_HTML.png)
whose mirror symmetry property reads \(E_1({{\overline{z}}}) = \overline{E_1(z)}\), see [1, p. 229, Eq. 5.1.13]. Obviously, we consider here the principal value of the integral when \(z \ne 0\), consult [23, p. 150, Eq. 6.2.1]. Moreover, in the Mathematica package the exponential integral is defined also as the principal value of the integral [23, p. 150, Eq. 6.2.5]
![](http://media.springernature.com/lw237/springer-static/image/art%3A10.1007%2Fs10998-022-00471-9/MediaObjects/10998_2022_471_Equ95_HTML.png)
However, the inter-connection \(E_1(x) = - {\mathrm{Ei}}(-x)\) holds true, see [23, p. 150, Eq. 6.2.6].
Theorem 3.1
For all \(r>0\), the following series expansions hold:
where \(\Re [z]\) denotes the real part of \(z \in {\mathbb {C}}\).
Proof
Expanding the secant hyperbolic kernel in the integrand of (1.3), for all \(x>0\) we have
Let \({\mathscr {L}}_x[f](s)\) denote the Laplace transform of a suitable function f with respect to the input variable x of the output variable s. By the expansion (3.3), the integral (1.3) becomes a series of Laplace transforms which reads
Next, we need the related Laplace integral property [1, p. 230, Eq. 5.1.28]
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 \(s = 2\pi (n+1)\) 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
therefore
Again, by the mirror symmetry of the exponential integral \(E_1(z)\), inserting \(s = \pi (2n+1)\), we conclude that
The rest is obvious. \(\square \)
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 \(n\ge 0\) 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, \(\lim _{n\rightarrow +\infty }u_n(r)=0\), and according to \(T_n(r)-S_n(r)=\frac{1}{2} u_{n+1}(r)\) 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.
Example 3.3
In Fig. 1 (left) we present the errors
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 \(n\le 5\). 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 \(r\in [0,1]\) 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 \({\mathcal {E}}_{S,n}(r)\) and presented in Fig. 2 (left), as well as the relative errors \({\mathcal {R}}_{S,n}(r)\) in the same figure (right).
Example 3.4
In the case of the alternating Mathieu series \({{\widetilde{S}}}(r)\) we study the auxiliary series
with the general term
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 \({\mathcal {R}}_{S,n}(r)\) (Fig. 2 (right)) and \({\mathcal {R}}_{{{\tilde{S}}},n}(r)\) (Fig. 4 (right)) for larger r and sufficiently large n (\(n=100\)), because of unstable calculations in such cases. Namely, the values S(r) and \({{\tilde{S}}}(r)\), 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.
4 The extended Mathieu series \({\mathbb {S}}_{\mu ,\nu }(r)\) and \(\widetilde{{\mathbb {S}}}_{\mu ,\nu }(r)\)
Motivated by (1.7), replacing there the kernel function \(Y_{-\frac{1}{2}}\) with the general Bessel function of the second kind of order \(\nu \), we introduce the extended Mathieu series S(r) and its alternating variant \({{\widetilde{S}}}(r)\) in the following forms:
where in both cases \(r>0,\,\mu >0\). Clearly \( {\mathbb {S}}_{\frac{5}{2}, -\frac{1}{2}}(r) = S(r)\) and \(\widetilde{{\mathbb {S}}}_{\frac{5}{2}, -\frac{1}{2}}(r) = {\widetilde{S}}(r)\).
Using the recurrence formula [30, p. 66, Eq. (1)]
we obtain the following recurrence formulae:
Theorem 4.1
If \(\mu ,\, \nu +1 >0\), \(n \in {\mathbb {N}}\) and \(\mu>|\nu |>0\), then
Moreover, if \(|\nu |<1\) and \(\mu +\nu +1>0\), then
where
Proof
Insert the binomial series expansion \(({\mathrm{e}}^x-1)^{-1} = \sum _{n \ge 1} {\mathrm{e}}^{-n x},\, x>0\) into (4.1). The legitimate integral-sum interchange which can be proved, e.g., by the dominated convergence theorem results in
Making use of the integral representation [30, p. 385, Eq. (4)] or, in other words, the Laplace–Mellin transform of the Bessel function \(Y_{\nu }\), i.e., \({\mathcal {L}}_a[x^{\mu -1}\,Y_{\nu }(b x)]\) and \({\mathcal {M}}_\mu [{\mathrm{e}}^{-a x}\,Y_{\nu }(b x)]\), respectively, we infer that
whose parameter space consists of \(\Re (\mu )>|\Re (\nu )|\) and \(\Re (a\pm ib)>0\), taken above \(a=n\) and \(b = r\), we conclude the first asserted formula. \(\square \)
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 \(p, q \in {\mathbb {C}}\) and \(-(p+q) \not \in {\mathbb {N}}\).
Theorem 4.2
If \(\mu ,\, \nu +1 >0\), \(n \in {\mathbb {N}}\), and \(\mu>|\nu |>0\), 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. 2]
whose parameter space consists of \(a>0,\,b>0,\,\Re (\mu )>|\Re (\nu )|\), for \(a=n\) and \(b = r\) we obtain the first asserted formula.
The derivation of the series expansion for \(\widetilde{{\mathbb {S}}}_{\mu ,\nu }(r)\) gives
Now, the path to the final formula is obvious. \(\square \)
5 Functional bounding inequalities
Recall the Gubler–Weber formula [30, p. 165, Eq. (5)]
which holds for \(\Re (z)>0\) and \(\nu >-1/2\). Splitting the \(\nu \)-domain into three disjoint intervals
Baricz et al. [3, pp. 957–958] obtained the functional bounding inequality for the real argument Neumann function \(Y_\nu (x)\) (see also [18, pp. 7–8], [11, p. 76]):
Theorem 5.1
If \(\mu ,\, \nu +\tfrac{1}{2} >0\), \(n \in {\mathbb {N}}\), and \(\mu>|\nu |>0\), then
Moreover, if \(\mu +\nu +1>0\), then
where
Proof
Starting with (4.1) and splitting the range of \(\nu \) into three disjoint intervals
and using the estimates (5.1), we conclude
where
and
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. \(\square \)
6 Extended Mathieu series in terms of the Riemann Zeta and Dirichlet Eta functions
The Bessel function of the second kind \(Y_\nu \) has two kinds of power series expansions depending on the nature of the order parameter. Firstly, when \(\nu =n \in {\mathbb {Z}}\), we have [1, p. 360, Eq. 9.1.11]
which immediately follows from (1.5) and (1.6). Here \(\psi \) is the digamma function defined by
For a noninteger order \(\nu \not \in {\mathbb {Z}}\) there exist several equivalent series representations; we work with the reformulated (1.6), viz.
Theorem 6.1
If \(\mu , r>0\) and \(n\in {\mathbb {N}}\), then
Proof
Consider (4.1) for \(\nu = n \in {\mathbb {N}}\). By the series (6.1) and by legitimate transformations we get
The first, third and fourth integrals are already known by virtue of (2.8), however, the second one is more challenging. Since
by the Mellin transform [16, p. 315, Eq. (9)]
and having in mind
setting \(p = \mu +2k+n\) and \(q = m+1\), we infer
Finally, applying (2.8) and (6.4) to the expression (6.3), after certain transformations and reduction, we arrive at the statement. \(\square \)
Theorem 6.2
If \(\mu , r>0\) and \(n\in {\mathbb {N}}_0\), then
Proof
Applying the Mellin transform
for all integrals which we derive by the lines of the previous proof, we clearly deduce the claimed result. \(\Box \) \(\square \)
Now, we present the Riemann Zeta building blocks series presentation of the extended Mathieu \({\mathbb {S}}_{\mu ,\nu }(r)\) and Dirichlet Eta function terms for extended alternating Mathieu series \(\widetilde{{\mathbb {S}}}_{\mu ,\nu }(r)\) by using the noninteger \(\nu \) parameter case.
Theorem 6.3
For all \(\mu , r>0\) and for \(|\nu |<1\), if \(\mu \pm \nu >1\), then
Moreover, for \(\mu , r>0\) and for \(|\nu |<1\), if \(\mu \pm \nu >0\), then
Proof
We start again with the integral (4.1) when \(\nu \in (-1,1)\). The series representation (6.2) implies
which is equivalent to the stated formula. The proof of (6.5) is now straightforward. \(\square \)
7 Extending the Butzer–Flocke–Hauss (complete) Omega function \(\varOmega (z)\) via Neumann functions
The notation \(\varOmega (z), z \in {\mathbb {C}}\), stands for the so-called complete Butzer–Flocke–Hauss (BHF) Omega function introduced in [4, Definition 7.1], [5] in the form
It is the Hilbert transform \({\mathscr {H}}_1[{\mathrm{e}}^{-zx}](0)\) at zero of the 1-periodic function \(\left( {\mathrm{e}}^{-zx}\right) _1\) defined by the periodic extension of the exponential function \({\mathrm{e}}^{-zx}\), \(|x|< \tfrac{1}{2},\, z \in {\mathbb {C}}\), thus
![](http://media.springernature.com/lw285/springer-static/image/art%3A10.1007%2Fs10998-022-00471-9/MediaObjects/10998_2022_471_Equ96_HTML.png)
Another expressions for the complete BHF Omega function \(\varOmega (x)\) are given by Butzer et al. [6]:
while the real argument complete BHF \(\varOmega \) function’s integral form by Tomovski and Pogány reads [29, p. 10, Theorem 3.3]
![](http://media.springernature.com/lw308/springer-static/image/art%3A10.1007%2Fs10998-022-00471-9/MediaObjects/10998_2022_471_Equ97_HTML.png)
By extensions in the integrand of the Butzer–Flocke–Hauss Omega function which is intimately connected to the generalized Mathieu series (consult the extensive study by Butzer and Pogány [5]) we are faced with a new territory of ideas and series/integral conclusion upon the structure of these kinds of generalizations.
Inspired by (7.1), we can write
having in mind that \(\cos (z) = -\sqrt{\pi z/2}\, Y_{\frac{1}{2}}(z)\) implementing the Neumann function of the general order \(\nu \) instead of \(Y_{-\frac{1}{2}}\) in the kernel in the following way:
The parameter range derivation will be our first goal. In turn, recognizing that the same integral consist both \(\varOmega _{\mu , \nu }(x)\) and \({\mathbb {S}}_{\mu ,\nu }(r)\) in (7.2) and (4.2), respectively, we deduce the relation
Therefore the parameter spaces coincide for any \(x>0\).
Next, the power series form of the complete BHF \(\varOmega \) function whose coefficients are built by finite sums containing Dirichlet Eta function terms is reported in [5, p. 901, Theorem 5.4. (ii)]
which shows that \(\varOmega \) is intimately connected with the Eta function. In [5] the authors discussed the relations of the Mathieu-, and the alternating Mathieu series and its generalized variants from one, and the \(\varOmega (z)\) function from other side by the Taylor expansion of the Hilbert–Eisenstein series \(\mathfrak h_1(z)\) of the first order and the polygamma function \(\psi ^{(r)}\) of order r (see also [2]).
However, our recent considerations are developed in another direction, according to the series representation of the expanded complete BHF \(\varOmega _{\mu , \nu }(x)\) in terms of the Dirichlet Eta function. In turn, bearing in mind (7.3), the counterpart results valid for \(\varOmega _{\mu , \nu }(x)\) exposed in Eq. (2.7) of Theorem 2.1, Theorem 6.2 and finally in Eq. (6.5) of Theorem 6.3 turn out to be their immediate consequences. So, we leave the formulation of these functional bound results to the interested reader.
References
M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1972)
Á. Baricz, P. L. Butzer, T. K. Pogány, Alternating Mathieu series, Hilbert – Eisenstein series and their generalized Omega functions. In G. V. Milovanović, M. Th. Rassias (eds.), Analytic Number Theory, Approximation Theory, and Special Functions – In Honor of Hari M. Srivastava. (Springer, New York, 2014), 775–808
Á. Baricz, D. Jankov Maširević, T.K. Pogány, Integral representations for Neumann-type series of Bessel functions \(I_{\nu }\), \(Y_{\nu }\) and \(K_{\nu }\). Proc. Amer. Math. Soc. 140(3), 951–960 (2012)
P.L. Butzer, S. Flocke, M. Hauss, Euler functions \(E_\alpha (z)\) with complex \(\alpha \) and applications, in Approximation, Probability and Related Fields. ed. by G.A. Anastassiou, S.T. Rachev (Plenum Press, New York, 1994), pp. 127–150
P.L. Butzer, T.K. Pogány, A fresh approach to classical Eisenstein series and the newer Hilbert-Eisenstein series. Int. J. Number Theory 13(4), 885–911 (2017)
P.L. Butzer, T.K. Pogány, H.M. Srivastava, A linear ODE for the Omega function associated with the Euler function \(E_\alpha (z)\) and the Bernoulli function \(B_\alpha (z)\). Appl. Math. Lett. 19, 1073–1077 (2006)
P. Cerone, C.T. Lenard, On integral forms of generalized Mathieu series. JIPAM J. Inequal. Pure Appl. Math. 4(5), Article No. 100, 1–11 (2003)
Junesang Choi, R.K. Parmar, T.K. Pogány, Mathieu-type series built by \((p, q)\)-extended Gaussian hypergeometric function. Bull. Korean Math. Soc. 54(3), 789–797 (2017)
Junesang Choi, H.M. Srivastava, Mathieu series and associated sums involving the Zeta functions. Comput. Math. Appl. 59(2), 861–867 (2010)
J. Clunie, On Bose-Einstein functions. Proc. Phys. Soc. Sect. A. 67(7), 632–636 (1954)
Dj. Cvijović, T.K. Pogány, Second type Neumann series related to Nicholson’s and to Dixon–Ferrar formula. In V. Kravchenko, S. Sitnik, (eds.), Transmutation Operators and Applications, Chapter 4. (Birkhäuser Verlag, Springer Basel AC, 2020), 67–84
N. Elezović, H.M. Srivastava, Ž Tomovski, Integral representations and integral transforms of some families of Mathieu type series. Integral Transforms Spec. Functions 19(7), 481–495 (2008)
O. Emersleben, Über die Reihe \(\sum _{k=1}^{\infty }\frac{k}{(k^{2}+r^{2})^{2}}\). Math. Ann. 125, 165–171 (1952)
A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, vol. I (McGraw-Hill Book Company, New York, Toronto and London, 1953)
A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, vol. II (McGraw-Hill Book Company, New York, Toronto and London, 1953)
A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Tables of Integral Transforms, vol. I (McGraw-Hill Book Company, New York, Toronto and London, 1954)
I.S. Gradshteyn, I.M. Ryzhik, Tables of Integrals, Series, and Products. (Corrected and Enlarged edition prepared by A. Jeffrey), seventh edition. (Academic Press, New York, 2007)
D. Jankov Maširević, T.K. Pogány, Second type Neumann series of generalized Nicholson function. Results Mat. 75(1), Article No. 12, 14pp (2020)
L. Lewin, Polylogarithms and Associated Functions (North-Holland, New York, 1981)
E.L. Mathieu, Traité de Physique Mathématique VI-VII: Théorie de l’élasticité des corps solides (Gauthier-Villars, Paris, 1890)
G.V. Milovanović, Summation of series and Gaussian quadratures. In: R.V.M. Zahar, (ed.), Approximation and Computation ISNM Vol. 119, 459–475, (Birkhäuser, Basel–Boston– Berlin, 1994)
G.V. Milovanović, T.K. Pogány, New integral forms of generalized Mathieu series and related applications. Appl. Anal. Discrete Math. 7, 180–192 (2013)
F.W.J. Olver, D.W. Lozier, R.F. Boisvert, C.W. Clark (eds.), NIST Handbook of Mathematical Functions (Cambridge University Press, Cambridge, 2010)
R.K. Parmar, G.V. Milovanović, T.K. Pogány, Multi-parameter Mathieu, and alternating Mathieu series. Appl. Math. Comp. 400, Article ID 126099, 27 pp. (2021)
R.K. Parmar, T.K. Pogány, On Mathieu-type series for the unified Gaussian hypergeometric functions. Appl. Anal. Discrete Math. 14(1), 138–149 (2020)
T.K. Pogány, R.K. Parmar, On \(p\)-extended Mathieu series. Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 22(534), 107–117 (2018)
T.K. Pogány, H.M. Srivastava, Ž Tomovski, Some families of Mathieu \({\mathbf{a}}\)-series and alternating Mathieu \(\mathbf{a}\)-series. Appl. Math. Comput. 173(1), 69–108 (2006)
H.M. Srivastava, Junesang Choi, Zeta and q-Zeta Functions and Associated Series and Integrals (Elsevier Science, Publishers, Amsterdam, London and New York, 2012)
Ž Tomovski, T.K. Pogány, Integral expressions for Mathieu-type power series and for the Butzer-Flocke-Hauss \(\Omega \)-function. Fract. Calc. Appl. Anal. 14(4), 623–634 (2011)
G.N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University Press, London, 1922)
D.C. Wood, The computation of polylogarithms. Technical Report 15-92 (PS), Canterbury, UK: University of Kent Computing Laboratory, 19pp (1992)
Acknowledgements
The authors are grateful to the anonymous referee for useful suggestions which certainly improved the accuracy and clarity of the exposition.
The work of the first author is supported by the Mathematical Research Impact Centric Support (MATRICS), SERB, Department of Science & Technology (DST), India (File No. MTR/2019/001328). The work of the second author was supported in part by the Serbian Academy of Sciences and Arts \((\varPhi \)-96). The research activities of the third author have been supported in part by the University of Rijeka, Croatia, under the project uniri-pr-prirod-19-16.
Funding
Open access funding provided by óbuda University.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Parmar, R.K., Milovanović, G.V. & Pogány, T.K. Extension of Mathieu series and alternating Mathieu series involving the Neumann function \(Y_\nu \). Period Math Hung 86, 191–209 (2023). https://doi.org/10.1007/s10998-022-00471-9
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10998-022-00471-9
Keywords
- Mathieu and alternating Mathieu series
- Neumann function \(Y_\nu \)
- Euler–Abel transformation of series
- Exponential integral \(E_1\)
- Gubler–Weber formula
- Associated Legendre function of second kind
- Riemann Zeta function
- Dirichlet Eta function
- Polylogarithm
- Complete Butzer–Flocke–Hauss \(\varOmega \) function
- Functional bounding inequality