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Abstract: Abstract The main focus of this paper is to investigate some new properties of the incomplete Fox–Wright function. We derive an integral representation of the incomplete Fox–Wright function whose terms contain Fox’s H-function. As a direct consequence, it leads to some new results including inequalities, log-convexity and complete monotonicity for this function. Moreover, it yields interesting monotonicity involving the ratios of the incomplete Fox–Wright function. Furthermore, certain generating functions for the incomplete Fox–Wright function when their terms contain the Appell function of the first kind are established. Finally, by means of the generating functions obtained here, new summation formula for the incomplete Fox–Wright function in terms of the H-function of two variables is made. PubDate: 2022-05-06

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Abstract: Abstract A stream of new theta relations is obtained. They follow from the general Thomae formula, which is a new result giving expressions for theta derivatives (the zero values of the lowest non-vanishing derivatives of theta functions with singular half-period characteristics) in terms of branch points and the period matrix of a hyperelliptic Riemann surface. The new theta relations contain (i) linear relations on the vector space of first-order theta derivatives which are arranged in gradients, (ii) relations between second-order theta derivatives and symmetric bilinear forms on the vector space of the gradients, (iii) relations between third-order theta derivatives and symmetric trilinear forms on the vector space of the gradients and (iv) a conjecture regarding higher-order theta derivatives. It is shown how the Schottky identity (in the hyperelliptic case) is derived from the obtained relations. PubDate: 2022-05-06

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Abstract: Abstract Ramanujan’s trigonometric sum \(c_q(n)\) can be interpreted as a set of trigonometric moments of a finite measure concentrated at primitive qth roots of unity with equal masses. This gives rise to sets of corresponding polynomials orthogonal on the unit circle. We present explicit expressions of these polynomials for special values of q, e.g., when \(q=p\) or \(q=2p\) or \(q=p^k\) , where p is a prime number. We generalize this procedure taking the Kronecker polynomial instead of cyclotomic one. In this case, the moments are expressed as finite sums of \(c_q(n)\) with different q. PubDate: 2022-05-04

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Abstract: Abstract Quite recently, many authors have investigated and found vanishing coefficients in the arithmetic progressions of several q-series expansions. In this paper we prove some new results on vanishing coefficients for arithmetic progressions modulo 3 and 5. We also obtain some relations between the coefficients of infinite q-series expansions. PubDate: 2022-05-04

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Abstract: Abstract Robin’s criterion states that the Riemann hypothesis is true if and only if the inequality \(\sigma (n) < e^{\gamma } \times n \times \log \log n\) holds for all natural numbers \(n > 5040\) , where \(\sigma (n)\) is the sum-of-divisors function of n and \(\gamma \approx 0.57721\) is the Euler–Mascheroni constant. We show that the Robin inequality is true for all natural numbers \(n > 5040\) that are not divisible by some prime between 2 and 1771559. We prove that the Robin inequality holds when \(\frac{\pi ^{2}}{6} \times \log \log n' \le \log \log n\) for some \(n>5040\) where \(n'\) is the square free kernel of the natural number n. The possible smallest counterexample \(n > 5040\) of the Robin inequality implies that \(q_{m} > e^{31.018189471}\) , \(1 < \frac{(1 + \frac{1.2762}{\log q_{m}}) \times \log (2.82915040011)}{\log \log n}+ \frac{\log N_{m}}{\log n}\) , \((\log n)^{\beta } < 1.03352795481\times \log (N_{m})\) and \(n < (2.82915040011)^{m} \times N_{m}\) , where \(N_{m} = \prod _{i = 1}^{m} q_{i}\) is the primorial number of order m, \(q_{m}\) is the largest prime divisor of n and \(\beta = \prod _{i = 1}^{m} \frac{q_{i}^{a_{i}+1}}{q_{i}^{a_{i}+1}-1}\) when n is an Hardy–Ramanujan integer of the form \(\prod _{i=1}^{m} q_{i}^{a_{i}}\) . PubDate: 2022-05-03

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Abstract: Abstract Let \(\alpha >1\) be an irrational number. We establish asymptotic formulas for the number of partitions of n into summands and distinct summands, chosen from the Beatty sequence \((\lfloor \alpha m\rfloor )\) . This improves some results of Erdös and Richmond established in 1977. PubDate: 2022-05-03

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Abstract: Abstract For a positive integer \(\ell \) , let \(b_{\ell }(n)\) denote the number of \(\ell \) -regular partitions of a nonnegative integer n. Motivated by some recent conjectures of Keith and Zanello, we establish infinite families of congruences modulo 2 for \(b_3(n)\) and \(b_{21}(n)\) . We prove a specific case of a conjecture of Keith and Zanello on self-similarities of \(b_3(n)\) modulo 2. We next prove that the series \(\sum _{n=0}^{\infty }b_9(2n+1)q^n\) is lacunary modulo arbitrary powers of 2. We also prove that the series \(\sum _{n=0}^{\infty }b_9(4n)q^n\) is lacunary modulo 2. PubDate: 2022-05-03

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Abstract: Abstract In this paper, we give an estimate of the lower bounds of the least common multiple of \(a,a+b,\ldots ,a+kb\) for \((a,b)=1,k\in \textit{N}^+\) . Precisely, we prove that for any two coprime positive integers a and b, we have $$\begin{aligned} L_{a,b,k}\ge \prod \limits _{p\mid b}p^{\text {Ord}_{p}^{k!}}\frac{1}{k!}\prod \limits _{i=0}^{k}(a+ib), \end{aligned}$$ where \(L_{a,b,k}\) is the least common multiple of \(a,a+b,\ldots ,a+kb\) and \(\text {Ord}_p^{k!}\) denotes the least s for which \(p^s\mid k!\) but \(p^{s+1}\not \mid k!\) . In addition, we obtain a corollary that there is a number containing a prime divisor greater than k in the set \(\{a,a+b,\ldots ,a+kb\}\) for \((a,b)=1,b\ge 2\) . PubDate: 2022-05-01

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Abstract: Abstract We prove that given any \(\epsilon >0\) , a non-zero adelic Hilbert cusp form \({\mathbf {f}}\) of weight \(k=(k_1,k_2,\ldots ,k_n)\in ({\mathbb {Z}}_+)^n\) and square-free level \(\mathfrak {n}\) with Fourier coefficients \(C_{{\mathbf {f}}}(\mathfrak {m})\) , there exists a square-free integral ideal \(\mathfrak {m}\) with \(N(\mathfrak {m})\ll k_0^{3n+\epsilon }N(\mathfrak {n})^{\frac{6n^2+1}{2}+\epsilon }\) such that \(C_{{\mathbf {f}}}(\mathfrak {m})\ne 0\) . The implied constant depends on \(\epsilon , F\) . PubDate: 2022-05-01

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Abstract: Abstract The evaluation of determinants with Legendre symbol entries is a classical topic both in number theory and in linear algebra. Recently Sun posed some conjectures on this topic. In this paper we confirm some of them via Gauss sums and the matrix determinant lemma. PubDate: 2022-05-01

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Abstract: Abstract For an elliptic curve \(E/{\mathbb {Q}}\) , let \(a_p\) denote the trace of its Frobenius endomorphism over \({\mathbb {F}}_p\) , where p is a prime of good reduction for E. Hasse’s theorem asserts that \( a_p \le 2\sqrt{p}\) . In this paper we establish average asymptotics for primes p for which \(a_p \in \left( 2\sqrt{p} - f(p), 2\sqrt{p}\right) \) or \(a_p \in \left( c\sqrt{p} - f(p), c\sqrt{p}\right) \) , where \(f(x) = o(\sqrt{x})\) is a function satisfying mild growth conditions and \(0< c < 2\) is a constant. PubDate: 2022-05-01

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Abstract: Abstract The alternating descent statistic on permutations was introduced by Chebikin as a variant of the descent statistic. We show that the alternating descent polynomials on permutations, called alternating Eulerian polynomials, are unimodal via a five-term recurrence relation. We also find a quadratic recursion for the alternating major index q-analog of the alternating Eulerian polynomials. As an interesting application of this quadratic recursion, we show that \((1+q)^{\lfloor n/2\rfloor }\) divides \(\sum _{\pi \in {{\mathfrak {S}}}_n}q^{\mathrm{altmaj}(\pi )}\) , where \({{\mathfrak {S}}}_n\) is the set of all permutations of \(\{1,2,\ldots ,n\}\) and \(\mathrm{altmaj}(\pi )\) is the alternating major index of \(\pi \) . This leads us to discover a q-analog of \(n!=2^{\ell }m\) , m odd, using the statistic of alternating major index. Moreover, we study the \(\gamma \) -vectors of the alternating Eulerian polynomials by using these two recursions and the cd-index. Further intriguing conjectures are formulated, which indicate that the alternating descent statistic deserves more work. PubDate: 2022-05-01

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Abstract: Abstract Inspired by Andrews and Merca’s recent work on the number of even parts over all partitions into distinct parts, we introduce a new kind of Beck type identities, which can be viewed as the dual form of Beck’s original conjectures. Based on a refinement of a special case of Franklin’s theorem, we obtain two interesting identities in great generality, corresponding to Yang’s generalization of Beck’s conjectures. Both analytic and combinatorial proofs of these results are provided. PubDate: 2022-05-01

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Abstract: Abstract Let \(\{d_{n}^{(v)}\}_{n\ge 0}\) be the sequence of generalized derangement numbers, where v is a nonnegative integer. In this paper, we mainly study the log-balancedness of \(\{d_{n}^{(v)}\}_{n\ge 0}\) , where \(v\ge 1\) . We prove that \(\{d_{n}^{(1)}\}_{n\ge 1}\) and \(\{d_{n}^{(v)}\}_{n\ge 0}\) ( \(v\ge 2\) ) are log-balanced. In addition, we discuss the log-balancedness of some sequences involving \(d_{n}^{(v)}\) . For example, we show that \(\{d_{n+1}^{(1)}-d_{n}^{(1)}\}_{n\ge 1}\) is log-balanced. PubDate: 2022-05-01

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Abstract: Abstract Heim and Neuhauser investigated some polynomials related to the Dedekind function. They proved the log-concavity of these polynomials and conjectured that they have only real zeros. We prove this conjecture, and deduce some identities for Fibonacci numbers and determine the modes of another sequence related to Fibonacci polynomials. PubDate: 2022-05-01

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Abstract: Abstract Recently, Mc Laughlin proved some results on vanishing coefficients in the series expansions of certain infinite q-products for arithmetic progressions modulo 5, modulo 7 and modulo 11 by grouping the results into several families. In this paper, we prove some new results on vanishing coefficients for arithmetic progressions modulo 7, which are not listed by Mc Laughlin. For example, we prove that if \(t \in \{1,2,3\}\) and the sequence \(\{A_n\}\) is defined by \(\sum _{n=0}^{\infty }A_nq^n := (-q^t,-q^{7-t};q^7)_{\infty }(q^{7-2t},q^{7+2t};q^{14})^3_{\infty },\) then \(A_{7n+4t}=A_{7n+6t^2+4t}=0\) for all n. Also, we prove four families of results with negative signs for arithmetic progressions modulo 7 classified by Mc Laughlin. PubDate: 2022-05-01

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Abstract: Abstract We study fixed points in integer partitions viewed, respectively, as weakly increasing or weakly decreasing structures. A fixed point is a point with value i in position i. We also study matching points in weakly decreasing partitions. These are defined as positions where the partition and its reverse have the same size parts. From the generating functions, we also obtain asymptotic estimates as \(n\rightarrow \infty \) of some of the above statistics. PubDate: 2022-05-01

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Abstract: Abstract Let \([n]=(1-q^n)/(1-q)\) denote the q-integer and \(\Phi _n(q)\) the nth cyclotomic polynomial in q. Recently, Guo and Schlosser provided two conjectures: For any odd integer \(n>3\) , modulo \([n]\Phi _n(q)(1-aq^n)(a-q^n)\) , $$\begin{aligned} \sum _{k=0}^{(n+1)/2}[4k+1]\frac{(aq^{-1};q^2)_k(q^{-1}/a;q^2)_k(q;q^2)_k^2}{(aq^4;q^2)_k(q^4/a;q^2)_k(q^2;q^2)_k^2}q^{4k} \equiv 0, \end{aligned}$$ and modulo \(\Phi _n(q)^2(1-aq^n)(a-q^n)\) , $$\begin{aligned} \sum _{k=0}^{(n+1)/2}[4k+1]\frac{(aq^{-1};q^2)_k(q^{-1}/a;q^2)_k(q^{-1};q^2)_k(q;q^2)_k}{(aq^4;q^2)_k(q^4/a;q^2)_k(q^4;q^2)_k(q^2;q^2)_k}q^{6k} \equiv 0, \end{aligned}$$ where \((a;q)_k=(1-a)(1-aq)\cdots (1-aq^{k-1})\) . In this paper, we confirm these two conjectures and further give their generalizations involving two free parameters. Our proof uses Guo and Zudilin’s ‘creative microscoping’ method and the Chinese remainder theorem for coprime polynomials. PubDate: 2022-05-01

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Abstract: Abstract It is well known that for all \(n\ge 1\) the number \(n+1\) is a divisor of the central binomial coefficient \({2n\atopwithdelims ()n}\) . Since the nth central binomial coefficient equals the number of lattice paths from (0, 0) to (n, n) by unit steps north or east, a natural question is whether there is a way to partition these paths into sets of \(n+1\) paths or \(n+1\) equinumerous sets of paths. The Chung–Feller theorem gives an elegant answer to this question. We pose and deliver an answer to the analogous question for \(2n-1\) , another divisor of \({2n\atopwithdelims ()n}\) . We then show our main result follows from a more general observation regarding binomial coefficients \({n\atopwithdelims ()k}\) with n and k relatively prime. A discussion of the case where n and k are not relatively prime is also given, highlighting the limitations of our methods. Finally, we come full circle and give a novel interpretation of the Catalan numbers. PubDate: 2022-05-01

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Abstract: Abstract In this paper, we focus on the generalized Marcum function of the second kind of order \(\nu >0\) , defined by $$\begin{aligned} R_{\nu }(a,b)=\frac{c_{a,\nu }}{a^{\nu -1}} \int _b ^ {\infty } t^{\nu } e^{-\frac{t^2+a^2}{2}}K_{\nu -1}(at)\mathrm{d}t, \end{aligned}$$ where \(a>0, b\ge 0,\) \(K_{\nu }\) stands for the modified Bessel function of the second kind, and \(c_{a,\nu }\) is a constant depending on a and \(\nu \) such that \(R_{\nu }(a,0)=1.\) Our aim is to find some new tight bounds for the generalized Marcum function of the second kind and compare them with the existing bounds. In order to deduce these bounds, we include the monotonicity properties of various functions containing modified Bessel functions of the second kind as our main tools. Moreover, we demonstrate that our bounds in some sense are the best possible ones. PubDate: 2022-05-01