Recent zbMATH articles in MSC 33Bhttps://www.zbmath.org/atom/cc/33B2021-04-16T16:22:00+00:00WerkzeugA comprehensible form of the product of two Gaussian \(Q\) functions and its usefulness in \(\kappa-\mu\) shadowed fading distribution.https://www.zbmath.org/1456.940252021-04-16T16:22:00+00:00"Sadhwani, Dharmendra"https://www.zbmath.org/authors/?q=ai:sadhwani.dharmendra"Yadav, Ram Narayan"https://www.zbmath.org/authors/?q=ai:yadav.ram-narayanSummary: \textit{G. K. Karagiannidis} and \textit{A. S. Lioumpas} [``An improved approximation for the Gaussian \(Q\)-function'', IEEE Commun. Lett. 11, No. 8, 644--646 (2007)], contrary to other approximations, proposed a more accurate approximation of the Gaussian \(Q\) function and its integer powers, for all the positive arguments. However, when it is used to compute the symbol error probability (SEP) of various coherent digital modulation techniques over parametric fading distributions like the \(\kappa-\mu\) shadowed fading, it loses its analytical tractability. In this paper, using Taylor series approximation of the exponential function, we comprehend the approximation of interest (with accuracy intact) which facilitates in the simplification of the key integrals used in the SEP computation of digital modulation techniques over \(\kappa-\mu\) shadowed fading distribution. Apart from various applications, the significance of the \(\kappa-\mu\) shadowed fading statistics lies in the fact that it unifies most of the popular fading models like one sided Gaussian, Rayleigh, Nakagami-\(m\), Nakagami-\(q\), Rician, Rician-shadowed, \(\eta-\mu\) and \(\kappa-\mu\), under one umbrella. In order to show the utility of the proposed work, the SEP of hexagonal-QAM is calculated over Rayleigh fading channel which is one of the widely used cases of the \(\kappa-\mu\) shadowed fading distribution. Monte Carlo simulations have also been carried out to justify the accuracy of the analysis.Identities and relations for Hermite-based Milne-Thomson polynomials associated with Fibonacci and Chebyshev polynomials.https://www.zbmath.org/1456.050072021-04-16T16:22:00+00:00"Kilar, Neslihan"https://www.zbmath.org/authors/?q=ai:kilar.neslihan"Simsek, Yilmaz"https://www.zbmath.org/authors/?q=ai:simsek.yilmazSummary: The aim of this paper is to give many new and interesting identities, relations, and combinatorial sums including the Hermite-based Milne-Thomson type polynomials, the Chebyshev polynomials, the Fibonacci-type polynomials, trigonometric type polynomials, the Fibonacci numbers, and the Lucas numbers. By using Wolfram Mathematica version 12.0, we give surfaces graphics and parametric plots for these polynomials and generating functions. Moreover, by applying partial derivative operators to these generating functions, some derivative formulas for these polynomials are obtained. Finally, suitable connections of these identities, formulas, and relations of this paper with those in earlier and future studies are designated in detail remarks and observations.Identities about level 2 Eisenstein series.https://www.zbmath.org/1456.110482021-04-16T16:22:00+00:00"Xu, Ce"https://www.zbmath.org/authors/?q=ai:xu.ceSummary: In this paper we consider certain classes of generalized level 2 Eisenstein series by simple differential calculations of trigonometric functions. In particular, we give four new transformation formulas for some level 2 Eisenstein series. We can find that these level 2 Eisenstein series are reducible to infinite series involving hyperbolic functions. Moreover, some interesting new examples are given.A sequence of modular forms associated with higher-order derivatives of Weierstrass-type functions.https://www.zbmath.org/1456.110502021-04-16T16:22:00+00:00"Aygunes, A. Ahmet"https://www.zbmath.org/authors/?q=ai:aygunes.aykut-ahmet"Simsek, Yılmaz"https://www.zbmath.org/authors/?q=ai:simsek.yilmaz"Srivastava, H. M."https://www.zbmath.org/authors/?q=ai:srivastava.hari-mohanSummary: In this article, we first determine a sequence \(\{f_n(\tau)\}_{n\in\mathbb N}\) of modular forms with weight
\[ 2^nk + 4(2^{n-1} - 1)\quad (n\in\mathbb N;\ k\in\mathbb N\backslash\{1\};\ \mathbb N:= \{1, 2, 3, \ldots\}). \]
We then present some applications of this sequence which are related to the Eisenstein series and the cusp forms. We also prove that higher-order derivatives of the Weierstrass type \(\wp_{2n}\)-functions are related to the above-mentioned sequence \(\{f_n(\tau)\}_{n\in\mathbb N}\) of modular forms.Exact lower and upper bounds on the incomplete gamma function.https://www.zbmath.org/1456.330022021-04-16T16:22:00+00:00"Pinelis, Iosif"https://www.zbmath.org/authors/?q=ai:pinelis.iosif-froimovichThe author proves first some new inequalities for the incomplete gamma function. Then he obtains some interesting lower and upper bounds, which are exact in a certain sense. Among others, a result by \textit{W. Gautschi} [J. Math. Phys., Mass. Inst. Techn. 38, 77--81 (1959; Zbl 0094.04104)] is improved.
Reviewer: József Sándor (Cluj-Napoca)Certain weighted averages of generalized Ramanujan sums.https://www.zbmath.org/1456.110072021-04-16T16:22:00+00:00"Namboothiri, K. Vishnu"https://www.zbmath.org/authors/?q=ai:namboothiri.k-vishnuEckford Cohen's generalization of the Ramanujan sums is defined as follows. If \(k,s\in {\mathbb N}\), \(j\in {\mathbb Z}\), then let
\[
c_k^{(s)}(j)= \sum_{\substack{1\le m\le k^s \\ (m,k^s)_s=1}} e^{2\pi i j m/k^s},
\]
where \((a,b)_s\) stands for the greatest common s-th power divisor of \(a\) and \(b\). If \(s=1\), then \(c_k^{(1)}(j)=c_k(j)\) are the classical Ramanujan sums. For every \(s\in {\mathbb N}\) one has
\[
c_k^{(s)}(j)= \sum_{\substack{d^s \mid j\\ d\mid k}} d^s \mu(k/d),
\]
where \(\mu\) is the Möbius function. Hence \(c_k^{(s)}(j)\) is integer valued and has properties similar to the Ramanujan sums \(c_k(j)\).
The reviewer [Ramanujan J. 35, No. 1, 149--156 (2014; Zbl 1322.11009)] established identities for certain weighted averages of the classical Ramanujan sums, with weights concerning
logarithms, values of arithmetic functions for gcd's, the Gamma function \(\Gamma\), the Bernoulli polynomials \(B_m\) and binomial coefficients.
In the present paper the author generalizes these identities to the Ramanujan sums \(c_k^{(s)}(j)\). He proves, among others, the following identities:
\[
\frac1{J_s(k)} \sum_{j=1}^{k^s} (\log \Gamma(j/k^s)) c_k^{(s)}(j) = \frac{s}{2} \sum_{p\mid k} \frac{\log p }{p^s-1} - \frac{\log 2\pi}{2},
\]
\[
\frac1{k^s} \sum_{j=0}^{k^s-1} B_m(j/k^s) c_k^{(s)}(j) = \frac{B_m}{k^{sm}} J_{sm}(k),
\]
where \(s,k,m\in {\mathbb N}\), \(J_s\) is the Jordan function of order \(s\) and \(B_m\) are the Bernoulli numbers.
Reviewer's remark: There are some misprints in the paper, also in the first two formulas of this review.
Reviewer: László Tóth (Pécs)Basic properties of incomplete Macdonald function with applications.https://www.zbmath.org/1456.330012021-04-16T16:22:00+00:00"Shu, Jian-Jun"https://www.zbmath.org/authors/?q=ai:shu.jianjun"Shastri, Kunal Krishnaraj"https://www.zbmath.org/authors/?q=ai:shastri.kunal-krishnarajThe purpose of this paper is to derive key properties of the Shu function \(S_{\nu}\), such as recurrence and differential relations and series and asymptotic expansions. A parabolic partial differential equation for \(S_{\nu}\) with a modified Bessel operator is derived. The proofs use integration by parts, incomplete gamma function, approximation of Macdonald function. Graphs for leading term approximations are shown.
Reviewer: Thomas Ernst (Uppsala)Self-avoiding walk on the complete graph.https://www.zbmath.org/1456.824512021-04-16T16:22:00+00:00"Slade, Gordon"https://www.zbmath.org/authors/?q=ai:slade.gordonThe phase transition for self-avoiding walks on the complete graph is studied.
It is found that the susceptibility, i.e., the generating function that counts
the number of self-avoiding walks according to their length, can be expressed
explicitly in terms of the incomplete gamma function. As long as the graph is
finite, the susceptibility is just a polynomial. Therefore, the asymptotic
behavior of the susceptibility is of interest, which is obtained from the
asymptotic behavior of the incomplete gamma function. The latter has a
transition in its asymptotic behavior, which in turn yields a phase transition
for the susceptibility. As main result, a critical scaling window for this
phase transition is identified, and the different behaviors are described that
occur below, above, and within the critical window.
Reviewer: Christoph Koutschan (Linz)