Some identities associated with mock theta functions $$\omega (q)$$ ω ( q ) and $$\nu (q)$$ ν ( q )

The Ramanujan Journal - Recently, Andrews, Dixit, and Yee defined two partition functions $$p_{\omega }(n)$$ and $$p_{\nu }(n)$$ that are related with Ramanujan’s mock theta functions...     

Congruences related to the Ramanujan/Watson mock theta functions $$\omega (q)$$ and $$\nu (q)$$ν(q)

We establish new connection formulae between Fibonacci polynomials and Chebyshev polynomials of the first and second kinds. These formulae are expressed in terms of certain values of hypergeometric functions of the type \(_2F_{1}\). Consequently, we obtain some new expressions for the celebrated Fibonacci numbers and their derivative sequences. Moreover, we evaluate some definite integrals involving products of Fibonacci and Chebyshev polynomials.     

$$I_0$$ and combinatorics at $$\lambda ^+$$λ+

We investigate the compatibility of \(I_0\) with various combinatorial principles at \(\lambda ^+\), which include the existence of \(\lambda ^+\)-Aronszajn trees, square principles at \(\lambda \), the existence of good scales at \(\lambda \), stationary reflections for subsets of \(\lambda ^{+}\), diamond principles at \(\lambda \) and the singular cardinal hypothesis at \(\lambda \). We also discuss whether these principles can hold in \(L(V_{\lambda +1})\).     

On hybrid moments of $$\Delta _2(x)$$ Δ 2 ( x ) and $$\Delta _3(x)$$ Δ 3 ( x )

The Ramanujan Journal - Let $$k\geqslant 2$$ be a fixed natural number and $$d_k(n)$$ denote the number of ways n can be written as a product of k positive integers. We use $$\Delta _k(x)$$ to...     

Dedekind $$\sigma $$ σ -complete $$\ell $$ ℓ -groups and Riesz spaces as varieties

Positivity - We prove that the category of Dedekind $$\sigma $$ -complete Riesz spaces is an infinitary variety, and we provide an explicit equational axiomatization. In fact, we show that finitely...     

On the Study of Processes of $$\sum (H)$$ and $$\sum _\mathrm{s}(H)$$∑s(H) Classes

In papers by Yor, a remarkable class \(({\varSigma })\) of submartingales is introduced, which, up to technicalities, are submartingales \((X_{t})_{t\ge 0}\) whose increasing process is carried by the times t such that \(X_{t}=0\). These submartingales have several applications in stochastic analysis: for example, the resolution of Skorokhod embedding problem, the study of Brownian local times and the study of zeros of continuous martingales. The submartingales of class \(({\varSigma })\) have been extensively studied in a series of articles by Nikeghbali (part of them in collaboration with Najnudel, some others with Cheridito and Platen). On the other hand, stochastic calculus has been extended to signed measures by Ruiz de Chavez (Le théorème de Paul Lévy pour des mesures signées. Séminaire de probabilités (Strasbourg). Springer, Berlin, 1984) and Beghdadi-Sakrani (Calcul stochastique pour les mesures signées. Séminaire de probabilités (Strasbourg). Springer, Berlin, 2003). In Eyi Obiang et al. (Stochastics 86(1):70–86, 2014), the authors of the present paper have extended the notion of submartingales of class \(({\varSigma })\) to the setting of Ruiz de Chavez (1984) and Beghdadi-Sakrani (2003), giving two different classes of stochastic processes named classes \(\sum (H)\) and \(\sum _\mathrm{s}(H)\) where from tools of the theory of stochastic calculus for signed measures, the authors provide general frameworks and methods for dealing with processes of these classes. In this work, we first give some formulas of multiplicative decomposition for processes of these classes. Afterward, we shall establish some representation results allowing to recover any process of one of these classes from its final value and the last time it visited the origin.     

On $$\gamma $$ γ - and local $$\gamma $$ γ -vectors of the interval subdivision

Journal of Algebraic Combinatorics - We show that the $$\gamma $$ -vector of the interval subdivision of a simplicial complex with a nonnegative and symmetric h-vector is nonnegative. In...     

Combinatorial proofs for identities related to generalizations of the mock theta functions $$\omega (q)$$ω(q) and $$\nu (q)$$ν(q)

The Ramanujan Journal - The two partition functions $$p_\omega (n)$$ and $$p_\nu (n)$$ were introduced by Andrews, Dixit and Yee, which are related to the third-order mock theta functions $$\omega...     

Stratification and $$\pi $$ π -cosupport: finite groups

Mathematische Zeitschrift - We introduce the notion of $$\pi $$ -cosupport as a new tool for the stable module category of a finite group scheme. In the case of a finite group, we use this to give...     

On Partition of Unities Generated by Entire Functions and Gabor Frames in $$ L^2({\mathbb R}^d) $$ and $$\ell ^2({\mathbb Z}^d)$$ℓ2(Zd)

We characterize the entire functions P of d variables, \(d\ge 2,\) for which the \({\mathbb Z}^d\)-translates of \(P\chi _{[0,N]^d}\) satisfy the partition of unity for some \(N\in \mathbb N.\) In contrast to the one-dimensional case, these entire functions are not necessarily periodic. In the case where P is a trigonometric polynomial, we characterize the maximal smoothness of \(P\chi _{[0,N]^d},\) as well as the function that achieves it. A number of especially attractive constructions are achieved, e.g., of trigonometric polynomials leading to any desired (finite) regularity for a fixed support size. As an application we obtain easy constructions of matrix-generated Gabor frames in \( L^2({\mathbb R}^d) ,\) with small support and high smoothness. By sampling this yields dual pairs of finite Gabor frames in \(\ell ^2({\mathbb Z}^d).\)     

Series representations for the Apery constant $$\zeta (3)$$ ζ ( 3 ) involving the values $$\zeta (2n)$$ ζ ( 2 n )

The Ramanujan Journal - In this note, using the well-known series representation for the Clausen function, we also provide some new representations of Apery’s constant $$\zeta (3)$$ . In...     

Congruences for the coefficients of the Gordon and McIntosh mock theta function $$\xi (q)$$ ξ ( q )

The Ramanujan Journal - Recently Gordon and McIntosh introduced the third order mock theta function $$\xi (q)$$ defined by $$\begin{aligned} \xi (q)=1+2\sum _{n=1}^{\infty...     

On $$ \mathcal{C} $$*-sets and decompositions of continuous and ηζ-continuous functions

The main purpose of this paper is to introduce the concepts of *-sets, *-continuous functions and to obtain new decompositions of continuous and ηζ-continuous functions. Moreover, properties of *-sets and some properties of -sets are discussed.      

The Dedekind $$\eta $$-function,a Hauptmodul for $$\Gamma _0(13)$$Γ0(13), and invariant theory

We give an explicit formula for the Hauptmodul \(\left( \frac{\eta (\tau )}{\eta (13 \tau )}\right) ^2\) of the level-13 Hecke modular group \(\Gamma _0(13)\) as a quotient of theta constants, together with some related explicit formulas. Similar results for primes \(p=2, 3, 5, 7\) (the other p for which \(\Gamma _0(p)\) has genus zero) are well known, and date back to Klein and Ramanujan. Moreover, we find an exotic modular equation, i.e., it has the same form as Ramanujan’s modular equation of degree 13, but with different kinds of modular parameterizations.     

A simpler proof of a Katsurada’s theorem and rapidly converging series for $$\zeta {(2n+1)}$$ ζ ( 2 n + 1 ) and $$\beta {(2n)}$$ β ( 2 n )

Annali di Matematica Pura ed Applicata (1923 -) - In a recent work on Euler-type formulae for even Dirichlet beta values, i.e., $$\beta {(2n)}$$ , I have derived an exact closed-form expression for...     

Evaluation of Series Involving the Product of the Tail of $${\zeta(k)}$$ and $${\zeta(k+1)}$$ζ(k+1)

We aim at evaluating the following class of series involving the product of the tail of two consecutive zeta function values

$$\sum\limits_{n=1}^{\infty}\left(\zeta(k)-1-\frac{1}{2^k}-\cdots-\frac{1}{n^k}\right)\left(\zeta(k+1)-1-\frac{1}{2^{k+1}}-\cdots-\frac{1}{n^{k+1}}\right),$$

where \({k\geq 2}\) is an integer. We show that the series can be expressed in terms of Riemann zeta function values and a special integral involving a polylogarithm function.

     

General fractional differential equations of order $$\alpha \in (1,2)$$ and Type $$\beta \in [0,1]$$β∈[0,1] in Banach spaces

In this paper, we are concerned with general fractional differential equations of order \(\alpha \in (1,2)\) and type \(\beta \in [0,1]\) in Banach spaces. We define and develop a theory of general fractional sine functions and show that they are essentiality equivalent to a general fractional resolvent. We use such theory to study the well-posedness of the above general fractional differential equations. An illustration example is presented.     

Relations among complementary and supplementary pairings of Saalschützian $${}_4F_3(1)$$ series

We investigate sums \(J(\vec {x})\) and \(L(\vec {x})\) of pairs of normalized Saalschützian \({}_4F_3(1)\) hypergeometric series, and develop a theory of relations among these J and L functions. The function \(L(\vec {x})\) has been studied extensively in the literature and has been shown to satisfy a number of two-term and three-term relations with respect to the variable \(\vec {x}\). More recent works have framed these relations in terms of Coxeter group actions on \(\vec {x}\) and have developed a similar theory of two-term and three-term relations for \(J(\vec {x})\). In this article, we derive “mixed” three-term relations, wherein any one of the L (respectively, J) functions arising in the above context may be expressed as a linear combination of two of the above J (respectively, L) functions. We show that, under the appropriate Coxeter group action, the resulting set of three-term relations (mixed and otherwise) among J and L functions partitions into eighteen orbits. We provide an explicit example of a relation from each orbit. We further classify the eighteen orbits into five types, with each type uniquely determined by the distances (under a certain natural metric) between the J and L functions in the relation. We show that the type of a relation dictates the complexity (in terms of both number of summands and number of factors in each summand) of the coefficients of the J and L functions therein.     

Weighted limits in an $$(\infty ,1)$$ ( ∞ , 1 ) -category

We introduce the notion of weighted limit in an arbitrary quasi-category, suitably generalizing ordinary limits in a quasi-category, and classical weighted limits in an ordinary category. This is accomplished by generalizing Joyal’s approach: we identify a meaningful construction for the quasi-category of weighted cones over a diagram in a quasi-category, whose terminal object is the weighted limit of the considered diagram. We then show that each weighted limit can be expressed as an ordinary limit. When the quasi-category arises as the homotopy coherent nerve of a category enriched over Kan complexes, we generalize an argument by Riehl-Verity to show that the weighted limit agrees with the homotopy weighted limit in the sense of enriched category theory, for which explicit constructions are available. When the quasi-category is complete, tensored and cotensored over the quasi-category of spaces, we discuss a possible comparison of our definition of weighted limit with the approach by Gepner-Haugseng-Nikolaus.

     

Quantitative weakly compact sets and Banach-Saks sets in $$\ell _1$$ ℓ 1

Archiv der Mathematik - In this paper, we show a quantitative version of the theorem stating that relatively weakly compact sets in $$\ell _1$$ coincide with those having the Banach-Saks property....