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.     

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...     

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...     

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...     

On the linear complexity of Hall’s sextic residue sequences over $${ GF}(q)$$

In this paper, we derive the linear complexity of Hall’s sextic residue sequences over the finite field of odd prime order. The order of the field is not equal to a period of the sequence. Our results show that Hall’s sextic residue sequences have high linear complexity over the finite field of odd order. Also we estimate the linear complexity of series of generalized sextic cyclotomic sequences. The linear complexity of these sequences is larger than half of the period.     

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.

     

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.     

(Non-)Distributivity of the product for $$\sigma $$ σ -algebras with respect to the intersection

Archiv der Mathematik - We study the validity of the distributivity equation $$\begin{aligned} ({\mathcal {A}}\otimes {\mathcal {F}})\cap ({\mathcal {A}}\otimes {\mathcal {G}}) ={\mathcal...     

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...     

Note on the number of zeros of $$\zeta ^{(k)}(s)$$ ζ ( k ) ( s )

The Ramanujan Journal - Assuming the Riemann hypothesis, we prove that $$\begin{aligned} N_k(T) = \frac{T}{2\pi }\log \frac{T}{4\pi e} + O_k\bigg (\frac{\log T}{\log \log T}\bigg ), \end{aligned}$$...     

On complemented subspaces of the spaces (l p ) N ∩l q (l q )

Summary In this paper we study the complemented subspaces of the spaces (l^p)^N∩l^q(l^q), with 1≤p<q≤∞ or q=0, thereby showing that if (l^p)^N∩l^q(l^q)=F⊕G then either F or G contains a complemented copy of the whole space. Entrata in Redazione il 19 aprile 1999. Ricevuta versione finale il 4 febbraio 2000. Research supported by the Italian MURST.     

On $${\mathcal {I}}_{

Tóth  János T.  Filip  Ferdinánd  Bukor  József  Zsilinszky  László 《Periodica Mathematica Hungarica》2021,82(2):125-135

A new proof of the existence of (q 2−q+1)-arcs inPG(2,q 2)

Journal of Geometry -     

The Gleason’s Problem on F(p,q,s) Type Spaces in the Unit Ball of $$\mathbf{C^{n}}$$

In this paper, the authors prove that the Gleason’s problem is solvable on the logarithm weight general function spaces F(p,q,s,k) in the unit ball \(B\subset \mathbf{C^{n}}\).     

On Ramanujan’s function k(q)=r(q)r 2(q 2)

In both his second and lost notebooks, Ramanujan introduced a function, related to the Rogers–Ramanujan continued fraction and its quadratic transformation, and listed several of its properties. We extend these results and develop a systematic theory.     

On the factor-web $$ {\boldsymbol{\bar{ W}}} $$ ( ρ, r,r) of the three-web W( r,r, r)

The notion of the factor-web [`(W)] \bar{W} (ρ, r, r) (1 ≤ ρ < r) is defined for the three-web W(r, r, r) formed on a 2r-dimensional differentiable manifold by three r-dimensional smooth foliations. Embedding of the factor-web in the initial web W(r, r, r) is constructed. This construction is a well-known geometric analog of the canonical extension of a Lie group of transformations to its parameter group.     

A remark on the size of πq (Sn)

Classical results of homotopy theory are used to prove that dim for all primes p. Simultaneously we obtain bounds on the order of the p-torsion subgroup of _q(Sⁿ).partially supported by Studienstiftung des deutschen Volkes     

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...     

Regular solutions to the Navier–Stokes equations with an initial data in $$ L(3,\infty )$$

We consider the Navier–Stokes initial boundary value problem in exterior domains \(\Omega \subset \mathbb {R}^n\), \(n\ge 3\). We assume that the initial data belongs to \({\mathbb {L}}(n,\infty )\) suitable subspace of \(L(n,\infty )\) Lorentz space. We are able to prove on an interval (0, T) the existence of a unique regular solution, global in time for small data. The solution enjoys some new estimates and a new approach to the proof is exhibited.     

Character tables of the association schemes coming from the action of G 2(q) on hyperplanes of type O 6 ε (q)

The character tables of the commutative association schemes coming from the action of the Chevalley group G ₂(q) on the set Ω _ε of hyperplanes of type O ₆ ^ε (q) in the seven dimensional orthogonal geometry over GF(q) related to the orthogonal group O ₇(q) are constructed by modifying the character tables of the association schemes obtained from the action of O ₇(q) on Ω _ε .