Recent zbMATH articles in MSC 33C60https://www.zbmath.org/atom/cc/33C602021-04-16T16:22:00+00:00WerkzeugHyperelliptic integrals modulo \(p\) and Cartier-Manin matrices.https://www.zbmath.org/1456.140362021-04-16T16:22:00+00:00"Varchenko, Alexander"https://www.zbmath.org/authors/?q=ai:varchenko.alexander-nSummary: The hypergeometric solutions of the KZ equations were constructed almost 30 years ago. The polynomial solutions of the KZ equations over the finite field \(\mathbb{F}_p\) with a prime number \(p\) of elements were constructed only recently. In this paper we consider an example of the KZ equations whose hypergeometric solutions are given by hyperelliptic integrals of genus \(g\). It is known that in this case the total \(2g\)-dimensional space of holomorphic (multivalued) solutions is given by the hyperelliptic integrals. We show that the recent construction of the polynomial solutions over the field \(\mathbb{F}_p\) in this case gives only a \(g\)-dimensional space of solutions, that is, a ``half'' of what the complex analytic construction gives. We also show that all the constructed polynomial solutions over the field \(\mathbb{F}_p\) can be obtained by reduction modulo \(p\) of a single distinguished hypergeometric solution. The corresponding formulas involve the entries of the Cartier-Manin matrix of the hyperelliptic curve. That situation is analogous to an example of the elliptic integral considered in the classical paper [\textit{Yu. I. Manin}, Izv. Akad. Nauk SSSR, Ser. Mat. 25, 153--172 (1961; Zbl 0102.27802)].Certain fractional integral and fractional derivative formulae with their image formulae involving generalized multi-index Mittag-Leffler function.https://www.zbmath.org/1456.260072021-04-16T16:22:00+00:00"Chand, Mehar"https://www.zbmath.org/authors/?q=ai:chand.mehar"Kasmaei, Hamed Daei"https://www.zbmath.org/authors/?q=ai:kasmaei.hamed-daei"Senol, Mehmet"https://www.zbmath.org/authors/?q=ai:senol.mehmetSummary: The main objective of this paper is to establish some image formulas by applying the Riemann-Liouville fractional derivative and integral operators to the product of generalized multiindex Mittag-Leffler function \(E^{\gamma,q}_{(\alpha_j,\beta_j)_m}(.)\). Some more image formulas are derived by applying integral transforms. The results obtained here are quite general in nature and capable of yielding a very large number of known and (presumably) new results.The irrationality measure of \(\pi\) is at most 7.103205334137\dots.https://www.zbmath.org/1456.111292021-04-16T16:22:00+00:00"Zeilberger, Doron"https://www.zbmath.org/authors/?q=ai:zeilberger.doron"Zudilin, Wadim"https://www.zbmath.org/authors/?q=ai:zudilin.wadimThe main result of this paper is that the irrationality measure exponent of the number \(\pi\) is less than \(7.103205334138\). The proof uses complex analysis, is based on clever calculating of special integral and is in the spirit of Salikov.
Reviewer: Jaroslav HanÄ¨l (Ostrava)Some results of fractional integral involving I-function and general class of polynomial.https://www.zbmath.org/1456.330122021-04-16T16:22:00+00:00"Tripathi, Bhupendra"https://www.zbmath.org/authors/?q=ai:tripathi.bhupendra"Sharma, Roshani"https://www.zbmath.org/authors/?q=ai:sharma.roshani"Sharma, C. K."https://www.zbmath.org/authors/?q=ai:sharma.chandra-kSummary: In the present paper, we have derived two multiplication theorems for I-function by using fractional integral formula \(I_{0,x}^{\alpha,\beta,\eta} f(x)\) and \(J_{0,\infty}^{\alpha,\beta,\eta} f(x)\) [\textit{P. K. Banerji} and \textit{S. Choudhary}, Proc. Natl. Acad. Sci. India, Sect. A 66, No. 3, 271--277 (1996; Zbl 1007.26501)]. Further, we established some applications of the theorem in the form of particular case.