Recent zbMATH articles in MSC 33https://www.zbmath.org/atom/cc/332022-05-16T20:40:13.078697ZWerkzeugA brief review of \(q\)-serieshttps://www.zbmath.org/1483.050132022-05-16T20:40:13.078697Z"Ismail, Mourad E. H."https://www.zbmath.org/authors/?q=ai:ismail.mourad-el-houssienyThe subject of \(q\)-series is very diverse. In the paper under review the author presents his philosophy on how to approach the subject by giving some examples. Central to the approach of the author is the use of a calculus for the Askey-Wilson operator and the identity theorem from analytic functions. The author builds his approach to \(q\)-series on the theory of the Askey-Wilson polynomials. One essential tool is the theory of \(q\)-Taylor series for the Askey-Wilson operator. Another ingredient is the Rodrigues-type-formulas for the Askey-Wilson polynomials
For the entire collection see [Zbl 1460.33001].
Reviewer: Zhi-Guo Liu (Shanghai)A family of \(q\)-hypergeometric congruences modulo the fourth power of a cyclotomic polynomialhttps://www.zbmath.org/1483.110012022-05-16T20:40:13.078697Z"Guo, Victor J. W."https://www.zbmath.org/authors/?q=ai:guo.victor-j-w"Schlosser, Michael J."https://www.zbmath.org/authors/?q=ai:schlosser.michael-jThe paper under review contains useful theorems. Taking Ramanujan's results for fast approximations of \(1/ \pi\), many authors established \(p\)-adic analogues of Ramanujan-type formulas. The authors of the present paper established a partial \(q\)-analogue of the supercongruence of the beautiful result of Long and Ramakrishna. In the present paper, they conjecture two none \(q\)-supercongruences and prove these by making use of Andrews multiseries extension of Watson's transformation and a Karlsson-Milton-type summation formula due to Gasper. The present paper includes beautiful and useful results which can be extended and used by the workers working in this field.
Reviewer: Vijay Yadav (Virar)On two supercongruences for sums of Apéry-like numbershttps://www.zbmath.org/1483.110022022-05-16T20:40:13.078697Z"Liu, Ji-Cai"https://www.zbmath.org/authors/?q=ai:liu.jicaiThe author proved that for any prime \(p>3\), we have
\[
\sum_{k=0}^{p-1}(2k+1)\frac{T_k}{4^k}\equiv p+\frac76p^4B_{p-3}\pmod{p^5}
\]
and
\[
\sum_{k=0}^{p-1}(2k+1)\frac{T_k}{(-4)^k}\equiv(-1)^{(p-1)/2}p+p^3E_{p-3}\pmod{p^4}.
\]
Those two congruences involving Apéry-like numbers were conjectured by Z.-H. Sun.
The author proves the above two congruences by using some combinatorial identities involving harmonic numbers and some related congruences.
Reviewer: Guo-Shuai Mao (Nanjing)Explicit evaluation formula for Ramanujan's singular moduli and Ramanujan-Selberg continued fractionshttps://www.zbmath.org/1483.110132022-05-16T20:40:13.078697Z"Prabhakaran, D. J."https://www.zbmath.org/authors/?q=ai:prabhakaran.devasirvatham-john"Ranjithkumar, K."https://www.zbmath.org/authors/?q=ai:ranjithkumar.kIt is well known that Ramanujan's general theta-function is defined by
\[
f(a,b)=\sum_{n=-\infty}^{+ \infty} a^{n(n+1)/2}\, b^{n(n-1)/2}, \qquad |ab|<1.
\]
Then let \(f(-q)=f(-q,-q^2)=(q;q^2)_{\infty},\) where \((a;q)_{\infty}= \prod_{n=0}^{\infty}(1-aq^n).\) There discusses the establishment of a few novel mixed modular equations using the theory of theta functions. Let \(S_1(q)\) and \(S_2(q)\) be the Ramanujan-Selberg continued fractions which are defined the following
\[
S_1(q) = \frac{q^{1/8}}{1} \genfrac{}{}{0pt}{0}{}{+}\frac{q}{1} \genfrac{}{}{0pt}{0}{}{+} \frac{q+q^2}{1} \genfrac{}{}{0pt}{0}{}{+} \frac{q^3}{1} \genfrac{}{}{0pt}{0}{}{+} \frac{q^2+q^4}{1} \genfrac{}{}{0pt}{0}{}{+}\genfrac{}{}{0pt}{0}{}{\dots}=\frac{q^{1/8}(-q^2;q^2)_{\infty}}{(-q;q^2)_{\infty}},
\]
\[
S_2(q) = \frac{q^{1/8}}{1} \genfrac{}{}{0pt}{0}{}{+}\frac{-q}{1} \genfrac{}{}{0pt}{0}{}{+} \frac{-q+q^2}{1} \genfrac{}{}{0pt}{0}{}{+} \frac{-q^3}{1} \genfrac{}{}{0pt}{0}{}{+} \frac{q^2+q^4}{1} \genfrac{}{}{0pt}{0}{}{+}\genfrac{}{}{0pt}{0}{}{\dots}=\frac{q^{1/8}(-q^2;q^2)_{\infty}}{(q;q^2)_{\infty}}, \quad |q|<1.
\]
The formulas for explicit evaluations of \(\alpha_{9n},\alpha_{n/9}\), \(S_1(q)\) and \(S_2(q)\) by modular equations establish in the paper. The several explicit values of Ramanujan's singular moduli and Ramanujan-Selberg continued fraction calculated.
Reviewer: Michael M. Pahirya (Mukachevo)Periodic ultradiscrete transformations of the plane with periods of 5, 7, 8, 9https://www.zbmath.org/1483.110192022-05-16T20:40:13.078697Z"Avdeeva, M. O."https://www.zbmath.org/authors/?q=ai:avdeeva.m-oSummary: \textit{V. A. Bykovskii} [Dokl. Math. 104, No. 2, 240--241 (2021; Zbl 07492936); translation from Dokl. Ross. Akad. Nauk, Mat. Inform. Protsessy Upr. 500, 23--25 (2021)] constructed three new periodic ultradiscrete transformations of the plane In addition to the two well-known. In his work, only the idea of proving these statements was proposed. We give a complete and detailed proof of them for sequences with periods 5, 7, 8, 9.Periodic ultradiscrete plane transformation with a period of 12https://www.zbmath.org/1483.110202022-05-16T20:40:13.078697Z"Monina, M. D."https://www.zbmath.org/authors/?q=ai:monina.m-dSummary: \textit{V. A. Bykovskii} [Dokl. Math. 104, No. 2, 240--241 (2021; Zbl 07492936); translation from Dokl. Ross. Akad. Nauk, Mat. Inform. Protsessy Upr. 500, 23--25 (2021)] constructed a new periodic ultradiscrete plane transformation with a period of 12. In his work only the idea of proving this periodicity was proposed. We provide a complete and detailed proof of this statement.A new property of Appell sequences and its applicationhttps://www.zbmath.org/1483.110352022-05-16T20:40:13.078697Z"Agoh, Takashi"https://www.zbmath.org/authors/?q=ai:agoh.takashiLet \(\left \{ A_{n}\left( x\right) \right \} _{n\geq 0}\) be a sequence of Appell polynomials, the author establishes the following property: for integers \(k,m\geq 0\) and variables \(x\) and \(y,\) one gets \(\left \{ A_{n}\left( x\right) \right \} _{n\geq 0}\) is an Appell sequence \textit{if and only if}
\[\left( A_{k}\left( x\right) +y\right) ^{m}=\left( A_{m}\left( x+y\right) -y\right) ^{k}.\]
This permits to the author to deduce several new recurrence relations for Bernoulli, Euler and Hermite polynomials. The author concludes the paper by establishing a bivariate Miki-type identity for Appell polynomials.
Reviewer: Hacène Belbachir (El Alia)The greatest lower bound of a Boros-Moll sequencehttps://www.zbmath.org/1483.110452022-05-16T20:40:13.078697Z"Pang, Sabrina X. M."https://www.zbmath.org/authors/?q=ai:pang.sabrina-x-m"Lv, Lun"https://www.zbmath.org/authors/?q=ai:lv.lun"Wang, Jiaxue"https://www.zbmath.org/authors/?q=ai:wang.jiaxueSummary: The Boros-Moll polynomials \(P_m(a)\) arise in the evaluation of a quartic integral. In the past few years, there has been some remarkable research on the properties of the Boros-Moll coefficients. \textit{W. Y. C. Chen} and \textit{C. C. Y. Gu} [Proc. Am. Math. Soc. 137, No. 12, 3991--3998 (2009; Zbl 1175.05019)] gave a lower bound of the sequence \(\{d^2_i(m)/d_{i-1}(m)d_{i+1}(m)\}\) for \(m\geq2\), which is a stronger result than the log-concavity of the sequence \(\{d_i(m)\}\). In this paper, we give the greatest lower bound for the sequence \(\{d^2_i(m)/d_{i-1}(m)d_{i+1}(m)\}\).Dilogarithm identities for solutions to Pell's equation in terms of continued fraction convergentshttps://www.zbmath.org/1483.110502022-05-16T20:40:13.078697Z"Bridgeman, Martin"https://www.zbmath.org/authors/?q=ai:bridgeman.martin-jSummary: We describe a new connection between the dilogarithm function and the solutions of Pell's equation \(x^2-ny^2=\pm 1\). For each solution \(x, y\) to Pell's equation, we obtain a dilogarithm identity whose terms are given by the continued fraction expansion of the associated unit \(x+y\sqrt{n}\in\mathbb{Z}[\sqrt{n}]\). We further show that Ramanujan's dilogarithm value-identities correspond to an identity for the regular ideal hyperbolic hexagon.New approach to Somos's Dedekind eta-function identities of level 6https://www.zbmath.org/1483.110792022-05-16T20:40:13.078697Z"Radha, D. Anu"https://www.zbmath.org/authors/?q=ai:radha.d-anu"Kumar, B. R. Srivatsa"https://www.zbmath.org/authors/?q=ai:kumar.belakavadi-radhakrishna-srivatsa"Shruthi"https://www.zbmath.org/authors/?q=ai:shruthi.shruthiSummary: In the present work, we prove few new Dedekind eta-function identities of level 6 discovered by Somos in two different methods. Also during this process, we give an alternate method to Somos's Dedekind eta-function identities of level 6 proved by B. R. Srivatsa Kumar et. al. As an application of this, we establish colored partition identities.The cubic moment of automorphic \(L\)-functions in the weight aspecthttps://www.zbmath.org/1483.111002022-05-16T20:40:13.078697Z"Frolenkov, Dmitry"https://www.zbmath.org/authors/?q=ai:frolenkov.dmitrii-aSummary: We prove an explicit formula for the cubic moment of central values of automorphic \(L\)-functions associated to primitive cusp forms of level one and large weight. The resulting explicit formula contains the main term predicted by the random matrix theory conjectures, while the error term is expressed as the fourth moment of the Riemann zeta function weighted by the \({}_3F_2\) hypergeometric function. As a corollary, we derive a new upper bound for the cubic moment improving the previous result of Peng. Furthermore, we obtain a new subconvexity estimate for automorphic \(L\)-functions in the weight aspect.Spherical functions and local densities on the space of \(p\)-adic quaternion Hermitian formshttps://www.zbmath.org/1483.111102022-05-16T20:40:13.078697Z"Hironaka, Yumiko"https://www.zbmath.org/authors/?q=ai:hironaka.yumikoOn algebraic values of Weierstrass \(\sigma \)-functionshttps://www.zbmath.org/1483.111522022-05-16T20:40:13.078697Z"Boxall, Gareth"https://www.zbmath.org/authors/?q=ai:boxall.gareth|boxall.gareth-j"Chalebgwa, Taboka"https://www.zbmath.org/authors/?q=ai:chalebgwa.taboka-prince"Jones, Gareth A."https://www.zbmath.org/authors/?q=ai:jones.gareth-aSummary: Suppose that \(\Omega\) is a lattice in the complex plane and let \(\sigma\) be the corresponding Weierstrass \(\sigma \)-function. Assume that the point \(\tau\) associated with \(\Omega\) in the standard fundamental domain has imaginary part at most 1.9. Assuming that \(\Omega\) has algebraic invariants \(g_2\), \(g_3\) we show that a bound of the form \(c d^m (\log H)^n\) holds for the number of algebraic points of height at most \(H\) and degree at most \(d\) lying on the graph of \(\sigma \). To prove this we apply results by Masser and Besson. What is perhaps surprising is that we are able to establish such a bound for the whole graph, rather than some restriction. We prove a similar result when, instead of \(g_2\), \(g_3\), the lattice points are algebraic. For this we naturally exclude those \((z,\sigma(z))\) for which \(z\in\Omega \).Trigonometric sums through Ramanujan's theory of theta functionshttps://www.zbmath.org/1483.111682022-05-16T20:40:13.078697Z"Harshitha, K. N."https://www.zbmath.org/authors/?q=ai:harshitha.k-n"Vasuki, K. R."https://www.zbmath.org/authors/?q=ai:vasuki.kaliyur-ranganna"Yathirajsharma, M. V."https://www.zbmath.org/authors/?q=ai:yathirajsharma.m-vSummary: The mathematics literature contains many generalized trigonometric sums which are evaluated through contour integration methods, algebraic methods or through discrete Fourier analysis methods. The purpose of this paper is to show how Ramanujan's theory of theta functions can be efficiently employed to evaluate certain generalized trigonometric sums. In the process, we obtain six interesting generalized trigonometric sums, that seem to be new.The twelfth moment of Dirichlet \(L\)-functions with smooth modulihttps://www.zbmath.org/1483.111812022-05-16T20:40:13.078697Z"Nunes, Ramon M."https://www.zbmath.org/authors/?q=ai:nunes.ramon-mSummary: We prove an analogue of \textit{D. R. Heath-Brown}'s bound on the 12th moment of the Riemann zeta function for Dirichlet \(L\)-functions with smooth moduli [Q. J. Math., Oxf. II. Ser. 29, 443--462 (1978; Zbl 0394.10020)].Evaluating log-tangent integrals via Euler sumshttps://www.zbmath.org/1483.111822022-05-16T20:40:13.078697Z"Sofo, Anthony"https://www.zbmath.org/authors/?q=ai:sofo.anthonySummary: An investigation into the representation of integrals involving the product of the logarithm and the arctan functions, reducing to log-tangent integrals, will be undertaken in this paper. We will show that in many cases these integrals take an explicit form involving the Riemann zeta function, the Dirichlet eta function, Dirichlet lambda function and many other special functions. Some examples illustrating the theorems will be detailed.Newton diagrams and the geometric degree of a polynomial mapping of \(\mathbf{C}^2\)https://www.zbmath.org/1483.140212022-05-16T20:40:13.078697Z"Masternak, Mateusz"https://www.zbmath.org/authors/?q=ai:masternak.mateuszSummary: Let \(H=(f, g) : \mathbb{C}^2\to\mathbb{C}^2\) be a polynomial mapping. We give a formula for the geometric degree of \(H\) in terms of the Newton diagrams of \(f\) and \(g\). We say that the mapping \(H\) is proper if for any compact set \(K\subset \mathbb{C}^2\) its preimage \(H^{-1}(K)\) is compact. In the paper a criterion of properness of the mapping \(H\) is also given.
For the entire collection see [Zbl 1481.26002].Weighted Hurwitz numbers, \(\tau\)-functions, and matrix integralshttps://www.zbmath.org/1483.140592022-05-16T20:40:13.078697Z"Harnad, J."https://www.zbmath.org/authors/?q=ai:harnad.johnThis paper deals with weighted Hurwitz Numbers, \(\tau\)-functions and matrix integrals. The basis elements spanning the Sato Grassmannian element corresponding to the KP \(\tau\)-function that serves as generating function for rationally weighted Hurwitz numbers are shown to be Meijer \(G\)-functions. Using their Mellin-Barnes integral representation the \(\tau\)-function, evaluated at the trace invariants of an externally coupled matrix, is expressed as a matrix integral. Using the Mellin-Barnes integral transform of an infinite product of \(G\)-functions, a similar matrix integral representation is given for the KP \(\tau\)-function that serves as generating function for quantum weighted Hurwitz numbers. This paper is organized as follows: Section 1 is devoted to Hurwitz numbers: classical and weighted, and Section 2 to hypergeometric \(\tau\)-functions as generating functions for weighted Hurwitz numbers. In theses Sections the author gives a brief review of this theory, together with two illustrative examples: rational and quantum weighted Hurwitz numbers. In Section 3, it is shown how evaluation of such \(\tau\)-functions at the trace invariants of a finite matrix may be expressed either as a Wronskian determinant or as a matrix integral.
For the entire collection see [Zbl 1471.81009].
Reviewer: Ahmed Lesfari (El Jadida)Multi-component universal character hierarchy and its polynomial tau-functionshttps://www.zbmath.org/1483.140602022-05-16T20:40:13.078697Z"Li, Chuanzhong"https://www.zbmath.org/authors/?q=ai:li.chuanzhong|li.chuanzhong.1The aim of this paper is to consider more general polynomial tau functions than universal characters of the UC hierarchy in [\textit{T. Tsuda}, Commun. Math. Phys. 248, No. 3, 501--526 (2004; Zbl 1233.37042)] which can be treated as a zero-mode of a certain generating tau function and the method used here is also a little different from what Tsuda have done in the above paper about UC hierarchy. Also the author generalizes these results of UC hierarchy to a multicomponent UC hierarchy. He finds that the polynomial tau-function in terms of universal characters of the universal character(UC) hierarchy which can be treated as a zero mode of an appropriate combinatorial generating function. After that, he defines a multi-component UC hierarchy and obtains polynomial tau-functions of the multi-component UC hierarchy. The paper is organized as follows : the first section is an introduction to the subject and formulation of the main result. Section 2 is devoted to symmetric functions. Section 3 deals with universal character and universal character hierarchy. Section 4 is devoted to polynomial tau-functions of the UC hierarchy and Section 5 to polynomial \(\tau\)-functions of the \(s\)-component UC hierarchy.
Reviewer: Ahmed Lesfari (El Jadida)Necessary and sufficient conditions for a difference constituted by four derivatives of a function involving trigamma function to be completely monotonichttps://www.zbmath.org/1483.330012022-05-16T20:40:13.078697Z"Qi, Feng"https://www.zbmath.org/authors/?q=ai:qi.fengSummary: In the paper, by virtue of convolution theorem for the Laplace transforms, Bernstein's theorem for completely monotonic functions, and other techniques, the authorfinds necessary and sufficient conditions for a difference constituted by four derivatives of a function involving trigamma function to be completely monotonic.New properties of the divided difference of psi and polygamma functionshttps://www.zbmath.org/1483.330022022-05-16T20:40:13.078697Z"Tian, Jing-Feng"https://www.zbmath.org/authors/?q=ai:tian.jingfeng"Yang, Zhen-Hang"https://www.zbmath.org/authors/?q=ai:yang.zhenhangThe authors define the functions
\[
\xi_{n}(x)=\frac{n \phi_{n}(x)}{\phi_{n+1}(x)}-x\quad\text{and}\quad\eta_{n}(x)=\left[\frac{\phi_{n}(x)}{(n-1) !}\right]^{-1 / n}-x
\]
and study some of their properties. Here \(-\phi_n(x)\) is the divided difference of the functions \(\psi_{n-1}(x+p)\) and \(\psi_{n-1}(x+q)\), where \(\psi_n=(-1)^{n-1}\psi^{(n)}\), and \(\psi^{(n)}\) stands for the \(n\)th polygamma function.
For example, it turns out that, under certain conditions,
\[
\lim _{n \rightarrow \infty} \xi_{n}(x)=\min \{p, q\}.
\]
It is also true, that \(\xi_n(x)\) is a convex function on \((-\min \{p, q\},\infty)\) iff \(p=q\). Similar statement holds true, mutatis mutandis, for \(\eta_n(x)\).
After proving these results, the paper contains applications of them in Section 4. These applications involve, for example, sharp inequalities for above defined \(\phi_n(x)\) maps. For instance, if \(n \in \mathbb{N}\), and \(|p-q| \gtrless 1\), then the inequalities
\[
x+\frac{p+q-1}{2} \gtrless \frac{n \phi_{n}(x)}{\phi_{n+1}(x)} \gtrless \frac{(n+1) \phi_{n+1}(x)}{\phi_{n+2}(x)} \gtrless x+\min \{p, q\}
\]
hold for every \(x\in(-\min \{p, q\},\infty)\). This yields, in the particular case when \(p=q=0\), an inequality for the \(\psi_n\) function:
\[
x-\frac{1}{2}<\frac{n \psi_{n}(x)}{\psi_{n+1}(x)}<\frac{(n+1) \psi_{n+1}(x)}{\psi_{n+2}(x)}<x
\]
for positive argument \(x\).
Another application is the presentation of the answer to Qi's and Agarwal's question about the monotonicity and convexity properties of the function
\[
f_{p, q ; \alpha, \beta}(x)=\left[\frac{\Gamma(x+p)}{\Gamma(x+q)}\right]^{\alpha /(p-q)}-\beta x.
\]
Reviewer: István Mező (Nanjing)Voros coefficients of the Gauss hypergeometric differential equation with a large parameterhttps://www.zbmath.org/1483.330032022-05-16T20:40:13.078697Z"Aoki, T."https://www.zbmath.org/authors/?q=ai:aoki.tosizumi|aoki.toshihiro|aoki.toru|aoki.takayuki|aoki.takuya|aoki.takafumi|aoki.toshihiko|aoki.toshiro|aoki.toshiaki|aoki.takashi|aoki.toshiki|aoki.takaaki|aoki.takayoshi|aoki.takeshi|aoki.takahiro|aoki.toshizumi|aoki.toshitaka|aoki.terumasa|aoki.takanori|aoki.takahira|aoki.toshiyuki"Takahashi, T."https://www.zbmath.org/authors/?q=ai:takahashi.tsuguo|takahashi.tomohiro|takahashi.takashi|takahashi.takenori|takahashi.tomoya|takahashi.toshinori|takahashi.toshitake|takahashi.tomoichi|takahashi.tsutomu|takahashi.tomokuni|takahashi.toshiaki|takahashi.tokiichiro|takahashi.takuya|takahashi.takayuki|takahashi.tomihiko|takahashi.tadashi|takahashi.tetsuya|takahashi.tadataka|takahashi.tadayasu|takahashi.takehito|takahashi.tohru|takahashi.toshio|takahashi.toru-t|takahashi.takao|takahashi.tsunero|takahashi.toyofumi|takahashi.toshimi|takahashi.tatsuji|takahashi.timothy-t|takahashi.teruo|takahashi.takuhiro|takahashi.taiki|takahashi.tomonori|takahashi.tsuyoshi|takahashi.tomokazu|takahashi.toshihiko|takahashi.toru|takahashi.takaaki|takahashi.takeshi.1|takahashi.tomoyuki|takahashi.tomohiko|takahashi.takeo|takahashi.toshimitsu|takahashi.tomoki|takahashi.tomo|takahashi.toshie"Tanda, M."https://www.zbmath.org/authors/?q=ai:tanda.mario|tanda.mikaSummary: The Voros coefficient of the Gauss hypergeometric differential equation with a large parameter is defined for the origin and its explicit form and the details of derivation are given.Sharp bounds for the weighted Hölder mean of the zero-balanced generalized complete elliptic integralshttps://www.zbmath.org/1483.330042022-05-16T20:40:13.078697Z"Zhao, Tie-Hong"https://www.zbmath.org/authors/?q=ai:zhao.tiehong"He, Zai-Yin"https://www.zbmath.org/authors/?q=ai:he.zaiyin"Chu, Yu-Ming"https://www.zbmath.org/authors/?q=ai:chu.yumingSummary: In the article, we present the best possible bounds for the weighted Hölder mean of the zero-balanced generalized complete elliptic integrals of the first and second kinds, which are the generalizations of previous results for complete \(p\)-elliptic integrals.A generalized sextic Freud weighthttps://www.zbmath.org/1483.330052022-05-16T20:40:13.078697Z"Clarkson, Peter A."https://www.zbmath.org/authors/?q=ai:clarkson.peter-a"Jordaan, Kerstin"https://www.zbmath.org/authors/?q=ai:jordaan.kerstinSummary: We discuss the recurrence coefficients of orthogonal polynomials with respect to a generalized sextic Freud weight
\[
\omega(x;t,\lambda)=|x|^{2\lambda+1} \operatorname{exp}(-x^6 + tx^2), \quad x \in \mathbb{R},
\]
with parameters \(\lambda > -1\) and \(t \in \mathbb{R}\). We show that the coefficients in these recurrence relations can be expressed in terms of Wronskians of generalized hypergeometric functions \(_1F_2(a_1;b_1;b_2;z)\). We derive a nonlinear discrete as well as a system of differential equations satisfied by the recurrence coefficients and use these to investigate their asymptotic behaviour. We conclude by highlighting a fascinating connection between generalized quartic, sextic, octic and decic Freud weights when expressing their first moments in terms of generalized hypergeometric functions.Certain properties of the Laguerre-Sheffer polynomialshttps://www.zbmath.org/1483.330062022-05-16T20:40:13.078697Z"Khan, Subuhi"https://www.zbmath.org/authors/?q=ai:khan.subuhi"Nahid, Tabinda"https://www.zbmath.org/authors/?q=ai:nahid.tabindaSummary: The present paper intends to study certain properties of the Laguerre-Sheffer polynomials utilizing matrix algebra. Some indispensable properties such as the recursive formulas, differential equations and identities for the Laguerre-Sheffer, Laguerre-associated Sheffer and Laguerre-Appell polynomials are established. This approach is mainly based upon the properties and relationships between the Pascal functional and Wronskian matrices. Examples providing the corresponding results for some hybrid special polynomials are derived which serve as a focal theme of the study.AFLT-type Selberg integralshttps://www.zbmath.org/1483.330072022-05-16T20:40:13.078697Z"Albion, Seamus P."https://www.zbmath.org/authors/?q=ai:albion.seamus-p"Rains, Eric M."https://www.zbmath.org/authors/?q=ai:rains.eric-m"Warnaar, S. Ole"https://www.zbmath.org/authors/?q=ai:warnaar.s-oleThe AFLT integral is an explicit evaluation of a Selberg-type integral of the product of two Jack polynomials obtained by Alba, Fateev, Litvinov and Tarnopolsky in their paper on the Alday-Gaiotto-Tachikawa (AGT) relation [\textit{V. A. Alba} et al., Lett. Math. Phys. 98, No. 1, 33--64 (2011; Zbl 1242.81119)]. The paper under this review gives several generalizations of the AFLT integral, containing (i) an \(A_n\) analogue of the AFLT integral, containing two Jack polynomials in the integrand; (ii) a partial generalization of (i), containing a product of \(n + 1\) Schur functions; (iii) an elliptic generalization, containing a pair of elliptic interpolation functions; (iv) an AFLT integral for Macdonald polynomials.
Reviewer: Shintaro Yanagida (Nagoya)New families of double hypergeometric series for constants involving \(1/\pi^2\)https://www.zbmath.org/1483.330082022-05-16T20:40:13.078697Z"Campbell, John Maxwell"https://www.zbmath.org/authors/?q=ai:campbell.john-maxwellSummary: We apply a Fourier-Legendre-based technique recently introduced by \textit{J. M. Campbell} et al. [J. Math. Anal. Appl. 479, No. 1, 90--121 (2019; Zbl 1423.33003)] to prove new rational double hypergeometric series formulas for expressions involving \(1/\pi^2\), especially the constant \(\zeta(3)/\pi^2\), which is of number-theoretic interest. Our techniques, applied in conjunction with Bonnet's recursion formula, give a powerful tool for evaluating double hypergeometric sums containing products of binomial coefficients, and yield many new transformation formulas for double hypergeometric series. The double series considered may be expressed as single sums involving the moments of elliptic-type integrals which have no known symbolic form.Some new mock theta functionshttps://www.zbmath.org/1483.330092022-05-16T20:40:13.078697Z"Cui, Su-Ping"https://www.zbmath.org/authors/?q=ai:cui.su-ping"Gu, Nancy S. S."https://www.zbmath.org/authors/?q=ai:gu.nancy-shan-shanSummary: Mock theta functions can be represented by Eulerian forms, Hecke-type double sums, Appell-Lerch sums, and Fourier coefficients of meromorphic Jacobi forms. In view of some \(q\)-series identities, we establish three two-parameter mock theta functions, and express them in terms of Appell-Lerch sums. In addition, we find three mock theta functions in Eulerian forms. Then in light of their Hecke-type double sums, we provide the Hecke-type double sums for some third order mock theta functions, and establish the relations between these functions and some classical mock theta functions.Some \(q\)-supercongruences modulo the square and cube of a cyclotomic polynomialhttps://www.zbmath.org/1483.330102022-05-16T20:40:13.078697Z"Guo, Victor J. W."https://www.zbmath.org/authors/?q=ai:guo.victor-j-w"Schlosser, Michael J."https://www.zbmath.org/authors/?q=ai:schlosser.michael-jSummary: Two \(q\)-supercongruences of truncated basic hypergeometric series containing two free parameters are established by employing specific identities for basic hypergeometric series. The results partly extend two \(q\)-supercongruences that were earlier conjectured by the same authors and involve \(q\)-supercongruences modulo the square and the cube of a cyclotomic polynomial. One of the newly proved \(q\)-supercongruences is even conjectured to hold modulo the fourth power of a cyclotomic polynomial.Absolutely monotonic functions involving the complete elliptic integrals of the first kind with applicationshttps://www.zbmath.org/1483.330112022-05-16T20:40:13.078697Z"Yang, Zhen-Hang"https://www.zbmath.org/authors/?q=ai:yang.zhenhang"Tian, Jing-Feng"https://www.zbmath.org/authors/?q=ai:tian.jingfengSummary: Let \(\mathcal{K}(r)\) be the complete elliptic integral of the first kind. In this paper, we prove that the function \(F_p(x)=(1-x)^p\exp\mathcal{K}(\sqrt{x})\) is absolutely monotonic on \((0,1)\) if and only if \(p\leq\pi/8\), and \(- F'_p(x)\) is absolutely monotonic on \((0,1)\) if and only if \(1/2\leq p\leq(\pi+4+\sqrt{16-\pi}/8\). This generalizes a known result and gives several new inequalities involving the complete elliptic integral of the first kind.Resolving singularities and monodromy reduction of Fuchsian connectionshttps://www.zbmath.org/1483.341212022-05-16T20:40:13.078697Z"Chiang, Yik-Man"https://www.zbmath.org/authors/?q=ai:chiang.yik-man"Ching, Avery"https://www.zbmath.org/authors/?q=ai:ching.avery"Tsang, Chiu-Yin"https://www.zbmath.org/authors/?q=ai:tsang.chiu-yinLet \(L(y)=0\) be the linear differential equation of Fuchsian class over \(\mathbb{C}(z)\). Under which conditions can solutions of this equation be expressed through hypergeometric functions? The answer to this question is important for many equations related to applied physical problems. Specifically for the Heun's differential equation \(\frac{d^{^{2}}y}{dz^{2}}+(\frac{\gamma}{z}+\frac{\delta}{z-1}+\frac{\epsilon}{z-t})\frac{dy}{dz}+\frac{\alpha\beta z-q}{z(z-1)(z-t)}=0\).
The general idea of solving such problems is to remove apparent singularities (see [\textit{M. A. Barkatou} and \textit{S. S. Maddah}, in: Proceedings of the 40th international symposium on symbolic and algebraic computation, ISSAC 2015, Bath, UK, July 6--9, 2015. New York, NY: Association for Computing Machinery (ACM). 53--60 (2015; Zbl 1346.68268)]). ``However, in order to overcome the ambiguity of the interplay between the local and global aspects of solutions of Fuchsian equations typically using classical language, it is the purpose of this article to apply sheaf theoretic language to study geometric aspects of Fuchsian connections where one of their singularities becomes apparent (i.e. being resolved).'' The proof of the effectiveness of the approach proposed by the authors may be their recover Takemura's eigenvalue inclusion theorem [\textit{K. Takemura}, J. Phys. A, Math. Theor. 45, No. 8, Article ID 085211, 14 p. (2012; Zbl 1247.34132)] and obtaining of new hypergeometric expansions of solutions to Heun's equations.
Reviewer: Mykola Grygorenko (Kyïv)Solutions of Painlevé II on real intervals: novel approximating sequenceshttps://www.zbmath.org/1483.341222022-05-16T20:40:13.078697Z"Bracken, Anthony J."https://www.zbmath.org/authors/?q=ai:bracken.anthony-jAn approximation method is proposed for a boundary value problem for the second Painlevé equation with Neumann boundary conditions
\[
y^{\prime\prime}(z) =2y(z)^3 +z y(z) +C, \quad y'(a)=0, \quad y'(b)=0, \quad (a<z<b).
\]
In the first step, a generalization of the second Painlevé equation is considered. The generalized equation on \(E(x)\) contains \(E(0), E(1)\) as nonlinear terms. In the expansion series \( E(x)=E_1(x) +E_2(x)+\cdots \), each \(E_n\) satisfies a nonhomogeneous Airy equation. By a suitable change of variables, a curious approximation series \(y_E^{(n)}\) is defined. \(y_E^{(n)}\) is a solution of a nonlinear equation with Neumann boundary conditions on an interval \([a_n, b_n\)], accompanied with a constant \(C_n\). When \(n \to \infty\), \(y_E^{(n)}\) converges to the solution \(y(z)\) and \(a_n, b_n\) and \(C_n\) also converges to \(a, b\) and \(C\), respectively, if the nonlinear term \(|y(z)|^3\) is small. A numerical example is also shown.
It is not clear why this method works well although the second Painlevé equation is closely related to the Airy function.
Reviewer: Yousuke Ohyama (Tokushima)Exact solutions of cubic-quintic modified Korteweg-de-Vries equationhttps://www.zbmath.org/1483.351892022-05-16T20:40:13.078697Z"Zemlyanukhin, Alexander I."https://www.zbmath.org/authors/?q=ai:zemlyanukhin.alexander-i"Bochkarev, Andrey V."https://www.zbmath.org/authors/?q=ai:bochkarev.andrey-vThe aim of this paper is to study a non-integrable modified Korteweg-de-Vries (mKdV) equation containing a combination of third and fifth degree nonlinear terms that simulate waves in a three layer fluid, as well as in spatially one-dimensional nonlinear elastic deformable systems. Using the Painlevé analysis [\textit{R. M. Conte} and \textit{M. Musette}, The Painlevé handbook. Dordrecht: Springer (2008; Zbl 1153.34002)], the authors study the analytical structure of equation, obtained from the mKdV 3-5 equation by transition to a traveling wave variable, build its solution, expressed in terms of the Weierstrass elliptic function [\textit{A. V. Porubov}, J. Phys. A, Math. Gen. 26, No. 17, L\, 797-L\, 800 (1993; Zbl 0803.35132); \textit{R. Racke}, Appl. Anal. 58, No. 1--2, 85--100 (1995; Zbl 0832.35097)], classify exact and approximate partial solitary-wave and periodic solutions and plot the corresponding graphs. It is established that mKdV equation passes the Painlevé test in a weak form. After the traveling wave transformation, this equation reduces to a generalized Weierstrass elliptic function equation, the right side of which is determined by a sixth order polynomial in the dependent variable. Determined by the structure of the polynomial roots, the general solution of the equation is expressed in terms of the Weierstrass elliptic function or its successive degenerations rational functions depending on the exponential functions of the traveling wave variable or directly on traveling wave variable. The classification of exact solitary wave and periodic solutions is carried out, and the ranges of parameters necessary for their physical feasibility are revealed. An approach is proposed for constructing approximate solitary wave and periodic solutions to generalized Weierstrass elliptic equation with a polynomial right hand side of high orders. This paper is organized as follows: The first section is an introduction to the subject. The second section deals with Painlevé analysis. The third and forth sections are devoted to periodic and soliton solutions. The fifth section deals with kink-shaped solution. The sixth section is devoted to approximate solution.
For the entire collection see [Zbl 1471.74003].
Reviewer: Ahmed Lesfari (El Jadida)On conditions of the completeness of some systems of Bessel functions in the space \(L^2 ((0;1); x^{2p} dx)\)https://www.zbmath.org/1483.420232022-05-16T20:40:13.078697Z"Khats, R. V."https://www.zbmath.org/authors/?q=ai:khats.r-vIn this paper the author gives necessary and sufficient conditions for the system \(\{x^{-p-1}\sqrt{x\rho_k}J_\nu(x\rho_k): k \in \mathbb{N}\}\) to be complete in the weighted space \(L^2((0,1), x^{2p} dx)\). Here \(J_\nu\) is the first kind Bessel function of index \(\nu \geq \frac{1}{2}\), \(p \in \mathbb{R}\) and \(\rho_k : k \in \mathbb{N}\) is an arbitrary sequence of distinct nonzero complex numbers.
The fact that \(\rho_k\) can be arbitrary had already been considered by \textit{B. V. Vynnyts'kyi} and \textit{R. V. Khats'} [Eurasian Math. J. 6, No. 1, 123--131 (2015; Zbl 1463.30015)]. In the present paper, he gives new conditions which depend only on properties of the \(\rho_k\).
Reviewer: Ursula Molter (Buenos Aires)Equivalence of \(K\)-functionals and modulus of smoothness generated by a generalized Jacobi-Dunkl transform on the real linehttps://www.zbmath.org/1483.440032022-05-16T20:40:13.078697Z"Daher, Radouan"https://www.zbmath.org/authors/?q=ai:daher.radouan"Tyr, Othman"https://www.zbmath.org/authors/?q=ai:tyr.othmanSummary: Using the generalized Jacobi-Dunkl translation, we prove the equivalence between modulus of smoothness and \(K\)-functional constructed by the Sobolev space corresponding to the Jacobi-Dunkl Laplacian operator.On the Fock kernel for the generalized Fock space and generalized hypergeometric serieshttps://www.zbmath.org/1483.460242022-05-16T20:40:13.078697Z"Park, Jong-Do"https://www.zbmath.org/authors/?q=ai:park.jong-doSummary: In this paper, we compute the reproducing kernel \(B_{m, \alpha}(z, w)\) for the generalized Fock space \(F_{m, \alpha}^2(\mathbb{C})\). The usual Fock space is the case when \(m=2\). We express the reproducing kernel in terms of a suitable hypergeometric series \({}_1 F_q\). In particular, we show that there is a close connection between \(B_{4, \alpha}(z, w)\) and the error function. We also obtain the closed forms of \(B_{m, \alpha}(z, w)\) when \(m=1,2/3,1/2\). Finally, we also prove that \(B_{m, \alpha}(z, z)\sim e^{\alpha |z|^m} |z|^{m-2}\) as \(|z|\longrightarrow\infty\).On the Bari basis properties of the root functions of non-self adjoint \(q\)-Sturm-Liouville problemshttps://www.zbmath.org/1483.470322022-05-16T20:40:13.078697Z"Allahverdiev, B. P."https://www.zbmath.org/authors/?q=ai:allahverdiev.bilender-pasaoglu"Tuna, H."https://www.zbmath.org/authors/?q=ai:tuna.huseyin|tuna.huseinSummary: This paper deals with the dissipative regular \(q\)-Sturm-Liouville problem. We prove that the system of root functions of this operator forms a Bari bases in the space \(L_q^2(I)\) by using the asymptotic behavior at infinity for its eigenvalues.Generalized Cesàro operators, fractional finite differences and gamma functionshttps://www.zbmath.org/1483.470652022-05-16T20:40:13.078697Z"Abadias, Luciano"https://www.zbmath.org/authors/?q=ai:abadias.luciano"Miana, Pedro J."https://www.zbmath.org/authors/?q=ai:miana.pedro-jSummary: In this paper, we present a complete spectral research of generalized Cesàro operators on Sobolev-Lebesgue sequence spaces. The main idea is to subordinate such operators to suitable \(C_0\)-semigroups on these sequence spaces. We introduce that family of sequence spaces using the fractional finite differences and we prove some structural properties similar to classical Lebesgue sequence spaces. In order to show the main results about fractional finite differences, we state equalities involving sums of quotients of Euler's Gamma functions. Finally, we display some graphical representations of the spectra of generalized Cesàro operators.Fractional-order Bessel wavelet functions for solving variable order fractional optimal control problems with estimation errorhttps://www.zbmath.org/1483.490062022-05-16T20:40:13.078697Z"Dehestani, Haniye"https://www.zbmath.org/authors/?q=ai:dehestani.haniye"Ordokhani, Yadollah"https://www.zbmath.org/authors/?q=ai:ordokhani.yadollah"Razzaghi, Mohsen"https://www.zbmath.org/authors/?q=ai:razzaghi.mohsenSummary: In the present paper, we apply the fractional-order Bessel wavelets (FBWs) for solving optimal control problems with variable-order (VO) fractional dynamical system. The VO fractional derivative operator is proposed in the sense of Caputo type. To solve the considered problem, the collocation method based on FBWFs, pseudo-operational matrix of VO fractional derivative and the dual operational matrix is proposed. In fact, we convert the problem with unknown coefficients in the constraint equations, performance index and conditions to an optimisation problem, by utilising the proposed method. Also, the convergence of the method with details is discussed. At last, to demonstrate the high precision of the numerical approach, we examine several examples.On a distinguished family of random variables and Painlevé equationshttps://www.zbmath.org/1483.600062022-05-16T20:40:13.078697Z"Assiotis, Theodoros"https://www.zbmath.org/authors/?q=ai:assiotis.theodoros"Bedert, Benjamin"https://www.zbmath.org/authors/?q=ai:bedert.benjamin"Gunes, Mustafa Alper"https://www.zbmath.org/authors/?q=ai:gunes.mustafa-alper"Soor, Arun"https://www.zbmath.org/authors/?q=ai:soor.arunSummary: A family of random variables \(\mathbf{X}(s)\), depending on a real parameter \(s>-\frac{1}{2} \), appears in the asymptotics of the joint moments of characteristic polynomials of random unitary matrices and their derivatives [\textit{T. Assiotis} et al., ``On the joint moments of the characteristic polynomials of random unitary matrices'', Preprint, \url{arXiv:2005.13961}], in the ergodic decomposition of the Hua-Pickrell measures [\textit{A. Borodin} and \textit{G. Olshanski}, Commun. Math. Phys. 223, No. 1, 87--123 (2001; Zbl 0987.60020); \textit{Y. Qiu}, Adv. Math. 308, 1209--1268 (2017; Zbl 1407.60011], and conjecturally in the asymptotics of the joint moments of Hardy's function and its derivative ([\textit{C. Hughes}, On the characteristic polynomial of a random unitary matrix and the Riemann zeta function. Heslington: University of York (PhD Thesis) (2001)] and [Assiotis et al., loc. cit.]). Our first main result establishes a connection between the characteristic function of \(\mathbf{X}(s)\) and the \(\sigma\)-Painlevé \(\text{III}^\prime\) equation in the full range of parameter values \(s>-\frac{1}{2} \). Our second main result gives the first explicit expression for the density and all the complex moments of the absolute value of \(\mathbf{X}(s)\) for integer values of \(s\). Finally, we establish an analogous connection to another special case of the \(\sigma \)-Painlevé \(\text{III}^\prime\) equation for the Laplace transform of the sum of the inverse points of the Bessel point process.On the continuous dual Hahn processhttps://www.zbmath.org/1483.601072022-05-16T20:40:13.078697Z"Bryc, Włodek"https://www.zbmath.org/authors/?q=ai:bryc.wlodzimierzThe author extends the continuous dual Hahn process \((\mathbb{T}_t)\) of \textit{I. Corwin} and \textit{A. Knizel} [``Stationary measure for the open KPZ equation'', Preprint, \url{arXiv:2103.12253}] from a finite time interval to the entire real line by taking a limit of a closely related Markov process \((T_t).\) The processes \((T_t) \) are characterized by conditional means and variances under bidirectional conditioning, and it is proved that continuous dual Hahn polynomials are orthogonal martingale polynomials for both processes.
Reviewer: Nasir N. Ganikhodjaev (Tashkent)A result on the Laplace transform associated with the sticky Brownian motion on an intervalhttps://www.zbmath.org/1483.601172022-05-16T20:40:13.078697Z"Song, Shiyu"https://www.zbmath.org/authors/?q=ai:song.shiyuLasso hyperinterpolation over general regionshttps://www.zbmath.org/1483.650292022-05-16T20:40:13.078697Z"An, Congpei"https://www.zbmath.org/authors/?q=ai:an.congpei"Wu, Hao-Ning"https://www.zbmath.org/authors/?q=ai:wu.hao-ningTranslation matrix elements for spherical Gauss-Laguerre basis functionshttps://www.zbmath.org/1483.650352022-05-16T20:40:13.078697Z"Prestin, Jürgen"https://www.zbmath.org/authors/?q=ai:prestin.jurgen"Wülker, Christian"https://www.zbmath.org/authors/?q=ai:wulker.christianSummary: Spherical Gauss-Laguerre (SGL) basis functions, i.e., normalized functions of the type \(L_{n-l-1}^{(l + 1/2)}(r^2) r^{l} Y_{lm}(\vartheta ,\varphi)\), \(|m| \le l < n \in{\mathbb{N}}\), constitute an orthonormal polynomial basis of the space \(L^{2}\) on \({\mathbb{R}}^{3}\) with radial Gaussian weight \(\exp (-r^{2})\). We have recently described reliable fast Fourier transforms for the SGL basis functions. The main application of the SGL basis functions and our fast algorithms is in solving certain three-dimensional rigid matching problems, where the center is prioritized over the periphery. For this purpose, so-called SGL translation matrix elements are required, which describe the spectral behavior of the SGL basis functions under translations. In this paper, we derive a closed-form expression of these translation matrix elements, allowing for a direct computation of these quantities in practice.Correction to: ``Translation matrix elements for spherical Gauss-Laguerre basis functions''https://www.zbmath.org/1483.650362022-05-16T20:40:13.078697Z"Prestin, Jürgen"https://www.zbmath.org/authors/?q=ai:prestin.jurgen"Wülker, Christian"https://www.zbmath.org/authors/?q=ai:wulker.christianCorrection to the authors' paper [ibid. 10, Paper No. 6, 16 p. (2019; Zbl 1483.65035), Formulae 3.6].Towards large-scale functional verification of universal quantum circuitshttps://www.zbmath.org/1483.810382022-05-16T20:40:13.078697Z"Amy, Matthew"https://www.zbmath.org/authors/?q=ai:amy.matthewSummary: We introduce a framework for the formal specification and verification of quantum circuits based on the Feynman path integral. Our formalism, built around exponential sums of polynomial functions, provides a structured and natural way of specifying quantum operations, particularly for quantum implementations of classical functions. Verification of circuits over all levels of the Clifford hierarchy with respect to either a specification or reference circuit is enabled by a novel rewrite system for exponential sums with free variables. Our algorithm is further shown to give a polynomial-time decision procedure for checking the equivalence of Clifford group circuits. We evaluate our methods by performing automated verification of optimized Clifford+\(T\) circuits with up to 100 qubits and thousands of \(T\) gates, as well as the functional verification of quantum algorithms using hundreds of qubits. Our experiments culminate in the automated verification of the Hidden Shift algorithm for a class of Boolean functions in a fraction of the time it has taken recent algorithms to simulate.
For the entire collection see [Zbl 1434.03017].A new approach to solve the Schrodinger equation with an anharmonic sextic potentialhttps://www.zbmath.org/1483.810632022-05-16T20:40:13.078697Z"Nanni, Luca"https://www.zbmath.org/authors/?q=ai:nanni.lucaSummary: In this study, the N-dimensional radial Schrodinger equation with an anharmonic sextic potential is solved by the extended Nikirov-Uranov method. We prove that the radial function can be factorised as the product between an exponential function and a polynomial function solution of the biconfluent Heun equation. The approach investigated in this article aims to be an alternative to other known methods of solving, as it has the advantage of dealing with simple, first-order differential and algebraic equations and avoiding numerous and laborious coordinate transformations and series expansions.Analytic derivation of the longitudinal proton structure function \(F_L(x, Q^2)\) and the reduced cross Section \(\sigma_r(x, Q^2)\) at the leading order and the next-to-leading order approximationshttps://www.zbmath.org/1483.810692022-05-16T20:40:13.078697Z"Zarrin, S."https://www.zbmath.org/authors/?q=ai:zarrin.saima|zarrin.sepideh"Dadfar, S."https://www.zbmath.org/authors/?q=ai:dadfar.sSummary: We present a set of formulas to extract the longitudinal proton structure function \(F_L(x, Q^2)\) and the reduced cross section \(\sigma_r(x, Q^2)\) by using Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations and Altarelli-Martinelli equation at the leading order (LO) and the next-to-leading order (NLO) approximations in perturbative quantum chromodynamics (QCD). These formulas are obtained by the Laplace transform method at the virtualities higher than \({Q_0^2}\). We show that, the obtained equations for \(F_L\) and \(\sigma_r\) depend only on the parton distribution functions (PDF's) at the initial scale \({Q_0^2}\). We obtain the corresponding numerical results in a range of the virtuality \({Q_0^2}\leq Q^2 \leq 800 GeV^2\) and Bjorken scale \(10^{- 4} \leq x \leq 1\) and compare them with the results achieved by MSTW2008 \textit{A. D. Martin} et al. [Eur. Phys. J. C, Part. Fields 63, No. 2, 189--285 (2009; Zbl 1369.81126)], WT \textit{C. D. White} and \textit{R. S. Thorne} Phy. Rev. D 75, No. 3 ,Article ID 034005, 28 p. (2007; \url{doi:10.1103/PhysRevD.75.034005})] and Dipole (b-Sat) model [\textit{H. Kowalski, L. Motyka} and \textit{G. Watt}, Phys. Rev. D 74, No. 7, Article ID 074016, 27 p. (2006; \url{doi:10.1103/PhysRevD.74.074016})] predictions and also with data released by the Hadron Electron Ring Accelerator (HERA). Our numerical results show an acceptable agreement with the deep inelastic scattering (DIS) experimental data throughout over a wide range of the Bjorken scale \(x\) and the virtuality \(Q^2\), and then can be applied in analyses of the Large Hadron Collider and Future Circular Collider projects.A set of \(q\)-coherent states for the Rogers-Szegő oscillatorhttps://www.zbmath.org/1483.810902022-05-16T20:40:13.078697Z"Mouayn, Zouhaïr"https://www.zbmath.org/authors/?q=ai:mouayn.zouhair"El Moize, Othmane"https://www.zbmath.org/authors/?q=ai:el-moize.othmaneSummary: We discuss a model of a \(q\)-harmonic oscillator based on Rogers-Szegő functions. We combine these functions with a class of \(q\)-analogs of complex Hermite polynomials to construct a new set of coherent states depending on a nonnegative integer parameter \(m\). Our construction leads to a new \(q\)-deformation of the \(m\)-true-polyanalytic Bargmann transform whose range defines a generalization of the Arik-Coon space. We also give an explicit formula for the reproducing kernel of this space. The obtained results may be exploited to define a \(q\)-deformation of the Ginibre-\(m\)-type process on the complex plane.Lie polynomials and a twistorial correspondence for amplitudeshttps://www.zbmath.org/1483.811022022-05-16T20:40:13.078697Z"Frost, Hadleigh"https://www.zbmath.org/authors/?q=ai:frost.hadleigh"Mason, Lionel"https://www.zbmath.org/authors/?q=ai:mason.lionel-jSummary: We review Lie polynomials as a mathematical framework that underpins the structure of the so-called double copy relationship between gauge and gravity theories (and a network of other theories besides). We explain how Lie polynomials naturally arise in the geometry and cohomology of \(\mathcal{M}_{0,n}\), the moduli space of \(n\) points on the Riemann sphere up to Mobiüs transformation. We introduce a twistorial correspondence between the cotangent bundle \(T^{\ast}_D\mathcal{M}_{0,n}\), the bundle of forms with logarithmic singularities on the divisor \(D\) as the twistor space, and \(\mathcal{K}_n\) the space of momentum invariants of \(n\) massless particles subject to momentum conservation as the analogue of space-time. This gives a natural framework for Cachazo He and Yuan (CHY) and ambitwistor-string formulae for scattering amplitudes of gauge and gravity theories as being the corresponding Penrose transform. In particular, we show that it gives a natural correspondence between CHY half-integrands and scattering forms, certain \(n-3\)-forms on \(\mathcal{K}_n\), introduced by Arkani-Hamed, Bai, He and Yan (ABHY). We also give a generalization and more invariant description of the associahedral \(n-3\)-planes in \(\mathcal{K}_n\) introduced by ABHY.Single-valued integration and superstring amplitudes in genus zerohttps://www.zbmath.org/1483.811172022-05-16T20:40:13.078697Z"Brown, Francis"https://www.zbmath.org/authors/?q=ai:brown.francis-c-s"Dupont, Clément"https://www.zbmath.org/authors/?q=ai:dupont.clementString amplitudes describe the interactions between states in string theory and are one of the most important observables. Indeed, they allow to reconstruct the low-energy effective action of string theory and to understand how the latter departs from usual local QFT. Moreover, these amplitudes display deep mathematical properties as they are obtained by integrating appropriate form over moduli spaces of Riemann surfaces. This paper studies open and closed string amplitudes with an arbitrary number of external states using the method of single-valued integration developed by the authors in a previous paper. The objective is to prove rigorously several properties of the amplitudes:
\begin{itemize}
\item[1.] Defining a ``canonical regularisation [string amplitudes] at tree level.''
\item[2.] Proving that the amplitudes ``admit a Laurent expansion in Mandelstam variables whose coefficients are multiple zeta values.''
\item[3.] Showing that ``closed string amplitudes are the single-valued projections of (motivic lifts of) open string amplitudes.''
\item[4.] Proving of the ``KLT formula expressing closed string amplitudes as quadratic expressions in open string amplitudes.''
\end{itemize}
Reviewer: Harold Erbin (Boston)Analytical angular solutions for the atom-diatom interaction potential in a basis set of products of two spherical harmonics: two approacheshttps://www.zbmath.org/1483.811632022-05-16T20:40:13.078697Z"Pawlak, Mariusz"https://www.zbmath.org/authors/?q=ai:pawlak.mariusz"Stachowiak, Marcin"https://www.zbmath.org/authors/?q=ai:stachowiak.marcinSummary: We present general analytical expressions for the matrix elements of the atom-diatom interaction potential, expanded in terms of Legendre polynomials, in a basis set of products of two spherical harmonics, especially significant to the recently developed adiabatic variational theory for cold molecular collision experiments [the first author et al., ``Adiabatic theory for anisotropic cold molecule collisions'', J. Chem. Phys. 143, No. 7, Article ID 074114, 5 p. (2015; \url{doi:10.1063/1.4928690}); ``Adiabatic variational theory for cold atom-molecule collisions: application to a metastable helium atom colliding with ortho- and para-hydrogen molecules'', J. Phys. Chem. A 121, No. 10, 2194--2198 (2017; \url{doi:10.1021/acs.jpca.6b13038})]. We used two approaches in our studies. The first involves the evaluation of the integral containing trigonometric functions with arbitrary powers. The second approach is based on the theorem of addition of spherical harmonics.