Narrow Results

The primary objective of this paper is to explore a novel subclass ${{\mathscr{H}}{𝒮}}_p(q,\alpha )$ of $p$-harmonic mappings, along with the associated subclass ${{\mathscr{H}}{𝒮}}_p^0(q,\alpha )$. We demonstrate that the mapping ${{\mathscr{H}}{𝒮}}_p(q,\alpha )$ is both univalent and sense-preserving within the unit disk $𝕌$. Furthermore, we determine the extreme points of ${{\mathscr{H}}{𝒮}}_p^0(q,\alpha )$ and establish that ${{\mathscr{H}}{𝒮}}_p(q,\alpha )\cap {{𝒯}}_p$ is defined. Additional results include the derivation of distortion bounds, the convolution condition, and the convex combination for this subclass. Finally, we examine the class-preserving integral operator and introduce a $q$-Jackson type integral operator.

In this paper we exploit Ruelle-type spectral functions and analyze the Verma module over Virasoro algebra, boson–fermion correspondence, the analytic torsion, the Chern–Simons and $\eta$ invariants, as well as the generating function associated to dimensions of the Hochschild homology of the crossed product $ℂ[S_n]⋉{{𝒜}}^{\otimes n}$ (${𝒜}$ is the $q$-Weyl algebra). After analyzing the Chern–Simons and $\eta$ invariants of Dirac operators by using irreducible $SU(n)$-flat connections on locally symmetric manifolds of nonpositive section curvature, we describe the exponential action for the Chern–Simons theory.

In this paper, we examine the Ruelle-type spectral functions ${ℛ}(s)$, which define an overall description of the content of the work. We investigate the Gopakumar–Vafa reformulation of the string partition functions, describe the $N=2$ Landau–Ginzburg model in terms of Ruelle-type spectral functions. Furthermore, we discuss the basic properties satisfied by elliptic genera in $N=2$ theories, construct the functional equations for ${ℛ}(s)$, and analyze the modular transformation laws for the elliptic genus of the Landau–Ginzburg model and study their properties in details.

In this work, we first introduce some new properties concerning the Fibonacci calculus. We then discuss the thermostatistics of gas models of two-parameter deformed oscillators, called bosonic and fermionic Fibonacci oscillators, in the thermodynamical limit. In this framework, we analyze the behavior of two-parameter deformed mean occupation numbers describing the Fibonacci-type bosonic and fermionic intermediate-statistics particles. A virial expansion of the equation of state for the bosonic Fibonacci oscillators’ gas model is obtained in both two and three dimensions, and the first five virial coefficients are derived in terms of the real independent deformation parameters $p$ and $q$. The effect of bosonic and fermionic $p$, $q$-deformation on the thermostatistical properties of Fibonacci-type $p$, $q$-boson and $p$, $q$-fermion gas models are also discussed. The results obtained in this work can be useful for investigating some exotic quasiparticle states encountered in condensed matter systems.

The aim of this paper is to derive the uncertainty principle which has implications in signal analysis and in quantum mechanics. First, we derive the definition of the wavelet transform in $q$-calculus by using some weight function. Certain properties like linearity, scaling, translation, etc. were discussed. Later on, to illustrate this integral transform several results were derived for the effectiveness and performance of the proposed method. Also, this paper surveys recent applications to establish generalized uncertainty principle.

The symmetric $q$-analysis is used to construct a type of minimum-uncertainty $q$-coherent states in the Fock representation space of the symmetric $q$-oscillator ∗-algebra with $0<q<1$. Then, its corresponding $q$-Hermite polynomials are derived by using the $q$-Bargmann–Fock realization of the symmetric $q$-oscillator algebra.

We use the recursion relations of the continuous $q$-Hermite polynomials and obtain the $q$-difference realizations of the ladder operators of a $q$-oscillator algebra in terms of the Askey–Wilson operator. For $q$-deformed coherent states associated with a disc in the radius $\frac{1}{\sqrt{1-q}}$, we obtain a compact form in $x$-representation by using the generating function of the continuous $q$-Hermite polynomials, too. In this way, we obtain a $q$-difference realization for the $q$-oscillator algebra in the finite interval $-\sqrt{\frac{2}{1-q}}<x<\sqrt{\frac{2}{1-q}}$ as a $q$-generalization of known differential formalism with respect to $x$ in the interval $-\infty <x<+\infty$ of the simple harmonic oscillator.

In this paper, we primarily consider a generalization of the fermionic $p$-adic $q$-integral on ${\mathbb{Z}}_p$ including the parameters $\alpha$ and $\beta$ and investigate its some basic properties. By means of the foregoing integral, we introduce two generalizations of $q$-Changhee polynomials and numbers as $q$-Changhee polynomials and numbers with weight $(\alpha ,\beta )$ and $q$-Changhee polynomials and numbers of second kind with weight $(\alpha ,\beta )$. For the mentioned polynomials, we obtain new and interesting relationships and identities including symmetric relation, recurrence relations and correlations associated with the weighted $q$-Euler polynomials, $\lambda$-Stirling numbers of the second kind and Stirling numbers of first and second kinds. Then, we discover multifarious relationships among the two types of weighted $q$-Changhee polynomials and $p$-adic gamma function. Also, we compute the weighted fermionic $p$-adic $q$-integral of the derivative of $p$-adic gamma function. Moreover, we give a novel representation for the $p$-adic Euler constant by means of the weighted $q$-Changhee polynomials and numbers. We finally provide a quirky explicit formula for $p$-adic Euler constant.

The aim of this paper is to define the operator of $q$-derivative based upon the Borel distribution and by using this operator, we familiarize a new subclass of $\beta$-uniformly starlike functions $\beta$-${𝒮}(\alpha ,q,\lambda ,\delta ).$ Further, we obtain coefficient estimates, distortion theorems, convex linear combinations and radii of close-to-convexity, starlikeness and convexity for functions $f\in \beta$-${𝒮}(\alpha ,q,\lambda ,\delta ).$ We also determine the second Hankel inequality for functions belonging to this subclass.

Based on $q$-Hypergeometric Series, a linear operator $H_s^r[{\alpha }_1;q]:{𝒜}\to {𝒜}$ is considered and involving this operator a class $S_q^{r,s}([{\alpha }_1];k,\alpha )$ of analytic functions is defined by using $q$-derivatives. As a special case, a class $H^q(a,b;c;k,\alpha )$ by involving $q$-analogue of Hohlov operator $I_{a,b,c}^q$ is defined. Coefficient inequality for a function $f$ to be in the class $S_q^{r,s}([{\alpha }_1];k,\alpha )$ is obtained. Further, in terms of subordination, an equivalent condition for a function $f$ to be in this class is given and using this equivalent class condition results on coefficient estimates including Fekete–Szegö inequality and a convolution result are obtained.