This paper deals with the study of global analysis following $(1,2)-$type system non-linear difference equations:\[ u_{n+1}=\frac{\alpha v_{n-1}}{\beta +\gamma v_{n}^{p}v_{n-2}^{q}},\ \ v_{n+1}=\frac{\alpha _{1}u_{n-1}}{\beta _{1}+\gamma _{1}u_{n}^{p}u_{n-2}^{q}},\ n=0,1,... \] where parameters $\alpha ,\beta ,\gamma ,\alpha _{1},\beta _{1},\gamma _{1,}p,q$ and initial conditions $u_{i},v_{i},$...

We use generating functions to express orthogonality relations in the form of q-beta integrals. The integrand of such a q-beta integral is then used as a weight function for a new set of orthogonal or biorthogonal functions. This method is applied to the continuous q-Hermite polynomials, the Al-Salam-Carlitz polynomials, and the polynomials of Szegő and leads naturally to the Al-Salam-Chihara p...

We use generating functions to express orthogonality relations in the form of q-beta integrals. The integrand of such a q-beta integral is then used as a weight function for a new set of orthogonal or biorthogonal functions. This method is applied to the continuous q-Hermite polynomials, the Al-Salam-Carlitz polynomials, and the polynomials of Szegő and leads naturally to the Al-Salam-Chihara p...

We have obtained a new summation formula for 2 ψ 2 bilateral basic hypergeometric series by the method of parameter augmentation and demonstrated its various uses leading to some development of etafunctions, q-gamma, and q-beta function identities.

Let $$p\in [1,\infty ]$$ , $$q\in )$$ $$s\in \mathbb {Z}_+:=\mathbb {N}\cup \{0\}$$ $$\alpha \in {R}$$ and $$\beta (0,1)$$ . In this article, the authors first find a reasonable version $$\widetilde{I}_{\beta }$$ of (generalized) fractional integral $$I_{\beta on special John–Nirenberg–Campanato space via congruent cubes, $$JN_{(p,q,s)_{\alpha }}^{\textrm{con}}(\mathbb {R}^n)$$ which coincides ...

The fundamental nature and origin of dark energy are one the premier mysteries theoretical physics. In General Relativity Theory, cosmological constant $\Lambda$ is simplest explanation for energy. On other hand, suffers from a delicate issue so-called fine-tuning problem. This motivates to modify spacetime geometry Einstein's GR. $f(Q)$ gravity recently proposed modified theory in which non-me...

We use the mirror coupling of Brownian motion to show that under a $\beta\in (0,1)$-dependent Kato type assumption (which is satisfied suitable $L^q$-assumption on electro-magnetic potential, where $q$ depends $\beta$ and dimension $d$) possibly nonsmooth corresponding magnetic Schr\"odinger semigroup in $\mathbb{R}$ has global $L^{p}$-to-$C^{0,\beta}$ H\"older smoothing property for all $p\in ...

Let $$G:\mathbb {R}\rightarrow \mathbb {R}$$ be a continuous function. In the first part of this paper, we investigate sufficient conditions on G such that $$\begin{aligned} \{G(f):f\in {\dot{K}}_{p,q}^{\alpha }F_{\beta }^{s}\}\subset {\dot{K}} _{p,q}^{\alpha }^{s} \end{aligned}$$ holds. Here $${\dot{K}}_{p,q}^{\alpha }^{s}$$ are Herz-type Triebel–Lizorkin spaces. These spaces unify and general...

In this paper, we have systematically discussed the existence of spherically symmetric wormhole solutions in framework $f(Q,\,T)$ gravity under two interesting non-commutative geometries such as Gaussian and Lorentzian distributions string theory. Also, to find solutions, consider models linear $f(Q,\,T)=\alpha\,Q+\beta\,T$ non-linear $f(Q,\,T)=Q+\lambda\,Q^2+\eta\,T$ our study. We obtained ana...