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Single-beam spectrally controlled (SBSC) two-dimensional (2D) Raman spectroscopy is a unique 2D vibrational measurement technique utilizing trains of short pulses that are generated from a single broadband pulse by pulse shaping. This approach overcomes the difficulty of 2D Raman spectroscopy in dealing with small-sig... |
Giant strides in ultrashort laser pulse technology have enabled real-time observation of dynamical processes in complex molecular systems. Specifically, the discovery of oscillatory transients in the two-dimensional electronic spectra of photosynthetic systems stimulated a number of theoretical investigations explorin... |
We investigate the construction of approximated exchange-correlation functionals by interpolating locally along the adiabatic connection between the weak- and the strong-coupling regimes, focussing on the effect of using approximate functionals for the strong-coupling energy densities. The gauge problem is avoided by ... |
A new approach for describing the effective electronic states of "atoms in compounds" to study the properties of molecules and condensed matter which are circumscribed by the operators heavily concentrated in atomic cores is proposed. Among the properties are hyperfine structure, space parity (P) and time reve... |
We investigate fractional-charge and fractional-spin errors in range-separated density-functional theory. Specifically, we consider the range-separated hybrid (RSH) method which combines long-range Hartree-Fock (HF) exchange with a short-range semilocal exchange-correlation density functional, and the RSH+MP2 method w... |
We present the formulation and implementation of the integral- and intermediate-screened coupled-cluster method (ISSCC). The IISCC method gives a simple and rigorous integral and intermediate screening (IIS) of the coupled-cluster method and will significantly reduces the scaling for all orders of the CC hierarchy exa... |
We show how Jarzynski relation can be exploited to analyze the nature of order-disorder and a bifurcation type dynamical transition in terms of a response function derived on the basis of work distribution over non-equilibrium paths between two thermalized states. The validity of the response function extends over lin... |
Fluorescence-detected two-dimensional electronic spectroscopy (F-2DES) projects the third-order non-linear polarization in a system as an excited electronic state population which is incoherently detected as fluorescence. Multiple variants of F-2DES have been developed. |
This article provides a review of recent work in the field of helium nanodroplet spectroscopy with emphasis on the dynamical aspects of the interactions between molecules in helium as well as their interaction with this unique quantum solvent. Emphasis is placed on experimental methods and studies introducing recent n... |
The recent advent of chirped-pulse FTMW technology has created a plethora of pure rotational spectra for molecules for which no vibrational information is known. The growing number of such spectra demands a way to build empirical potential energy surfaces for molecules, without relying on any vibrational measurements.... |
Time-resolved scattering experiments enable imaging of materials at the molecular scale with femtosecond time resolution. However, in disordered media they provide access to just one radial dimension thus limiting the study of orientational structure and dynamics. |
With the emergence of Transformer architectures and their powerful understanding of textual data, a new horizon has opened up to predict the molecular properties based on text description. While SMILES are the most common form of representation, they are lacking robustness, rich information and canonicity, which limit... |
To accelerate the thermal equilibrium sampling of multi-level quantum systems, the infinite swapping limit of a recently proposed multi-level ring polymer representation is investigated. In the infinite swapping limit, the ring polymer evolves according to an averaged Hamiltonian with respect to all possible surface i... |
The strongly-contracted variant of second order N -electron valence state perturbation theory (NEVPT2) is an efficient perturbative method to treat dynamic correlation without the problems of intruder states or level shifts, while the density matrix renormalization group (DMRG) provides the capability to tackle static ... |
The structure of thiolate-protected gold cluster Au38(SR)24 has not been determined experimentally and the best available signature is its measured optical spectrum. Using this signature and energetic stability as criteria and SCH3 for SR, we compare four candidate structures: two from others and two we obtained. |
Quasicentroid molecular dynamics (QCMD) is a path-integral method for approximating nuclear quantum effects in dynamics simulations, which has given promising results for gas- and condensed-phase water. Here, by simulating the infrared spectrum of gas-phase ammonia, we test the feasibility of extending QCMD beyond wat... |
We have experimentally determined the lateral registry and geometric structure of free-base porphine (2H-P) and copper-metalated porphine (Cu-P) adsorbed on Cu(111), by means of energy-scanned photoelectron diffraction (PhD), and compared the experimental results to density functional theory (DFT) calculations that inc... |
Tasker type III polar terminations of ionic crystals carry a net surface charge as well as a dipole moment and are fundamentally unstable. In contact with electrolytes, such polar surfaces can be stabilized by adsorption of counter ions from solution to form electric double layers (EDLs). |
Low-energy electron impact excitations of S_2 molecules are studied using the fixed-bond R-matrix method based on state-averaged complete active space SCF orbitals. Integral cross sections are calculated for elastic electron collision as well as impact excitation of the 7 lowest excited electronic states. |
We report pressure broadening coefficients for the 21 electric-dipole transitions between the eight lowest rotational levels of ortho-H$_2$O and para-H$_2$O molecules by collisions with He at temperatures from 20 to 120 K. These coefficients are derived from recently published experimental state-to-state rate coefficie... |
We compute the decay rates of emitters coupled to spheroidal nanoantennas made of gold, copper, silver, and aluminum. The spectral position of the localized surface plasmon-polariton resonance, the enhancement factors and the quantum efficiency are investigated as a function of the aspect ratio, background index and t... |
Controlling matter at the level of electrons using ultrafast laser sources represents an important challenge for science and technology. Recently we introduced a general laser control scheme (the Stark Control of Electrons at Interfaces or SCELI) based on the Stark effect that uses the subcycle structure of light to m... |
It is pointed out that the derivative of the energy density functional does not provide a valid local electronegativity measure, in spite of its appealing property of becoming constant for ground-state equilibrium systems. |
Results of a systematic investigation of the vapour-liquid equilibria (VLE) of 30 individual two-centre Lennard-Jones plus pointquadrupole model fluids (2CLJQ) are reported over a range of reduced quadrupolar momentum $0 \le Q^{*2} \le 4$ and of reduced elongation $0 \le L^* \le 0.8$. Temperatures investigated are fro... |
Ultrafast dynamics in chemical systems provide a unique access to fundamental processes at the molecular scale. A proper description of such systems is often very challenging because of the quantum nature of the problem. |
Super and hyperhalogens are a class of highly electronegative species whose electron affinities far exceed those of halogen atoms and are important to chemical industry as oxidizing agents, bio-catalysts, and building blocks of salts. Using Wade-Mingos rule well known for describing the stability of closo-boranes (BnH... |
A rotating particle in electrorheological (ER) fluid leads to a displacement of its polarization charges on the surface which relax towards the external applied field ${\bf E}_0$, resulting in a steady-state polarization at an angle with respect to ${\bf E}_0$. This dynamic effect has shown to affect the ER fluids pro... |
We present a low-complexity algorithm to calculate the correlation energy of periodic systems in second-order Møller-Plesset perturbation theory (MP2). In contrast to previous approximation-free MP2 codes, our implementation possesses a quartic scaling, $\mathcal O (N^4)$, with respect to the system size $N$ and offer... |
The bulk self diffusion of the ethylammonium cation and the nitrate anion can both be described by respective single diffusion coefficients. Diffusion of the anion is 1.7 times higher than that of the cation. |
Given the Caputo-type fractional differential equation $D^\alpha y(t) = f(t, y(t))$ with $\alpha \in (1, 2)$, we consider two distinct solutions $y_1, y_2 \in C[0,T]$ to this equation subject to different sets of initial conditions. In this framework, we discuss nontrivial upper and lower bounds for the difference $|y... |
A spectral parameter power series (SPPS) representation for the regular solution of the radial Dirac system with complex coefficients is obtained, as well as a SPPS representation for the (entire) characteristic function of the corresponding spectral problem on a finite interval. Based on the SPPS representation, a nu... |
Two classifications of second order ODE's cubic with respect to the first order derivative are compared in the first case of intermediate degeneration. The correspondence of vectorial, pseudovectorial, scalar, and pseudoscalar invariants of the equations in this case is established. |
In this paper, we give a two dimensional analogue of the Euler-MacLaurin summation formula. By using this formula, we obtain an integral representation of Weil's elliptic functions which was introduced in the book "Elliptic functions according to Eisenstein and Kronecker". |
The Cholesky factorization of the moment matrix is considered for discrete orthogonal polynomials of hypergeometrical type. We derive the Laguerre-Freud equations when the first moments of the weights are given by the ${}_1F_2$, ${}_2F_2$ and ${}_3F_2$ generalized hypergeometrical functions. |
Using the concept of $\mathcal{D}$-operator and the classical discrete family of dual Hahn, we construct orthogonal polynomials $(q_n)_n$ which are also eigenfunctions of higher order difference operators. |
In a recent work by Cruz-Uribe et al. was obtained that \[|\{x\in{\mathbb{R}^d}:w(x)|G(fw^{-1})(x)|>\alpha\}|\lesssim\frac{[w]_{A_1}^2}{\alpha}\int_{{\mathbb{R}^d}}|f|dx\] both in the matrix and scalar settings, where $G$ is either the Hardy-Littlewood maximal function or any Calderón-Zygmund operator. In this note... |
The concept of unique normal form is formulated in terms of a spectral sequence. As an illustration of this technique some results of Baider and Churchill concerning the normal form of the anharmonic oscillator are reproduced. |
A bilinear inequality of Geba, Greenleaf, Iosevich, Palsson, and Sawyer for the Fourier transform is shown to be equivalent to a simpler linear inequality, and the range of exponents is extended. Related mixed-norm inequalities are discussed. |
Given a sequence of polynomials $(p_n)_n$, an algebra of operators $\mathcal{A}$ acting in the linear space of polynomials and an operator $D_p\in \mathcal{A}$ with $D_p(p_n)=np_n$, we form a new sequence of polynomials $(q_n)_n$ by considering a linear combination of $m$ consecutive $p_n$: $q_n=p_n+\sum_{j=1}^m\beta_{... |
Consider the Gelfand pairs $(G_p,K_p):=(M_{p,q} \rtimes U_p,U_p)$ associated with motion groups over the fields $\mathbb F=\mathbb R,\mathbb C,\mathbb H$ with $p\geq q$ and fixed $q$ as well as the inductive limit $p\to\infty$,the Olshanski spherical pair $(G_\infty,K_\infty)$. We classify all Olshanski spherical func... |
Let $\Omega $ be any set of directions (unit vectors) on the plane. We study maximal operators defined by \md0 M_\Omega f(x)=\sup_{\delta >0, \omega \in \Omega} \frac{1}{2\delta}\int_{-\delta}^\delta |f(x+t\omega)|dt. |
We show that the Jacobi polynomials that are orthogonal on the unit circle (the Jacobi OPUC) are CMV bispectral. This means that the corresponding Laurent polynomials in the CMV basis satisfy two dual ordinary eigenvalue problems: a recurrence relation and a differential equation of Dunkl type. |
In this paper we prove some sharp weighted norm inequalities for the multi(sub)linear maximal function $\Mm$ introduced in \cite{LOPTT} and for multilinear Calderón-Zygmund operators. In particular we obtain a sharp mixed "$A_p-A_{\infty}$" bound for $\Mm$, some partial results related to a Buckley-type estima... |
The Bernstein polynomials with integer coefficients do not generally preserve monotonicity and convexity. We establish sufficient conditions under which they do. |
First we recall the notion of conxity and log-convexity for real-valued. Then we generalize the trick used by Artin in his famous paper on the Gamma function to find log-convex solutions to the functional equations f(x+1)=g(x)f(x). |
Consider a_1,a_2,...,a_n, arbitrary elements of R. We characterize those real functions f that decompose into the sum of a_j-periodic functions, i.e., f=f_1+...+f_n with D_{a_j}f(x):=f(x+a_j)-f(x)=0. We show that f has such a decomposition if and only if for all partitions to B_1, B_2,... |
In this note we are dealing with the problem of existence and asymptotic behavior of solutions for the non-resonant singularly perturbed linear Neumann boundary value problem \begin{eqnarray*} \epsilon y"+ky=f(t),\quad k>0,\quad 0<\epsilon<<1,\quad t\in\langle a,b\rangle \end{eqnarray*} \begin{equation*... |
The principal aim of this paper is to extend Birman's sequence of integral inequalities originally obtained in 1961, and containing Hardy's and Rellich's inequality as special cases, to a sequence of inequalities that incorporates power weights on either side and logarithmic refinements on the right-hand si... |
We present an elementary proof of a conjecture proposed by I. Rasa in 2017 which is an inequality involving Bernstein basis polynomials and convex functions. It was affirmed in positive by A. Komisarski and T. Rajba very recently by the use of stochastic convex orderings. |
This paper is a continuation of the recent paper "CMV biorthogonal Laurent polynomials: Christoffel formulas for Christoffel and Geronimus transformations" by the same authors. The behavior of quasidefinite sesquilinear forms for Laurent polynomials in the complex plane, characterized by bivariate linear funct... |
Lectures notes (in italian) of some arguments of classical analysis, with exercises. A particular emphasis to functional analysis and elementary operator algebra theory is given, by means of exercises and examples. |
The beta integral is applied to accelerate the hypergeometric function $2 F 1\left\{1, B; C ; w\right\}$ to derive new infinite series for constants such as $\pi$ and values of the gamma function. A compendium of new infinite series is given. |
In this note we will show a Calderón--Zygmund decomposition associated with a function $f\in L^1(\mathbb{T}^{\omega})$. The idea relies on an adaptation of a more general result by J. L. Rubio de Francia in the setting of locally compact groups. |
The movement of data (communication) between levels of a memory hierarchy, or between parallel processors on a network, can greatly dominate the cost of computation, so algorithms that minimize communication are of interest. Motivated by this, attainable lower bounds for the amount of communication required by algorit... |
We show that there exist infinitely many nontrivial choices of parameters of the single confluent Heun equation for which the three-term recurrence relations governing the expansions of the solutions in terms of the confluent hypergeometric functions 1F1 and 0F1 are reduced to two-term ones. In such cases the expansio... |
We study non-symmetric Jacobi polynomials of type $BC_1$ by means of vector-valued and matrix-valued orthogonal polynomials. The interpretation as matrix-valued orthogonal polynomials allows us to introduce shift operators for the non-symmetric Jacobi polynomials. |
These notes are connected to a "potpourri" topics class and deal with some special cases of norms of various objects which arise in classical analysis. |
Given a set of functions F={f_1,...,f_m} of L2(Rd), we study the problem of finding the shift-invariant space V with n generators {phi_1,...,phi_n} that is ``closest'' to the functions of F in the sense that V minimize the least square distance from the data F to V over the set of all shift-invariant spaces tha... |
A spectral parameter power series (SPPS) representation for solutions of Sturm-Liouville equations of the form $$(pu')'+qu=u\sum_{k=1}^{N}\lambda^{k}r_{k}$$ is obtained. It allows one to write a general solution of the equation as a power series in terms of the spectral parameter $\lambda$. |
In this short note we have proved an enhanced version of a theorem of Lorentz [1] and its generalization to the multivariate case which gives a non- uniform estimate of degree of approximation by a polynomial with positive coefficients. The performance of the approximation at the vertices of [0; 1]d is more precisely ... |
In the present paper, we introduce a Choldowsky type generalization of the q Favard-Szasz operators and obtain weighted statistical approximation properties of these operators. We also establish the rates of statistical convergence by means of the modulus of continuity and the Lipschitz type maximal function. |
In this article, the authors first introduce the Triebel-Lizorkin-type space $F_{p(\cdot),q(\cdot)}^{s(\cdot),\phi}(\mathbb R^n)$ with variable exponents, and establish its $\varphi$-transform characterization in the sense of Frazier and Jawerth, which further implies that this new scale of function spaces is well defi... |
An explicit family of polynomials on the unit ball $B^d$ of $\RR^d$ is constructed, so that it is an orthonormal family with respect to the inner product $$ <br>< f,g > <br>= \rho \int_{B^d}\nabla f(x)\cdot \nabla g(x) dx + \CL (fg), $$ where $\rho >0$, $\nabla$ is the gradient, and $\CL(fg)$ is either the inn... |
Using the duplication formulas of the elliptic trigonometric functions of Gosper, we deduce some new special values for the first two Jacobi theta functions. At the end of the paper, we show how is it possible to extend our arguments and deduce a wide variety of additional special values for the Jacobi thetas. |
We prove in this paper the following estimate for the maximal operator $T^*$ associated to the singular integral operator $T$: $ \|T^*f\|_{L^{1,\infty} (w)} \lesssim \frac{1}{\epsilon} <br>\int_{\mathbb{R}^n} |f |
In this paper, an inequality of Simpson type for quasi-convex mappings are proved. The constant in the classical Simpson's inequality is improved. |
We study the projections in vector spaces over finite fields. We prove finite fields analogues of the bounds on the dimensions of the exceptional sets for Euclidean projection mapping. |
In this short research note we obtain double definite integral expressions for the Kapteyn type series built by Kummer's $M$ (or confluent hypergeometric ${}_1F_1$) functions. These kind of series unify in natural way the similar fashion results for Neumann-, Schlömilch- and Kapteyn-Bessel series recently establis... |
When $G(z)$ is a power series in $z$, many authors now write `$[z^n] G(z)$' for the coefficient of $z^n$ in $G(z)$, using a notation introduced by Goulden and Jackson in [\GJ, p. 1]. More controversial, however, is the proposal of the same authors [\GJ, p. 160] to let `$[z^n/n!] G(z)$' denote the coefficient o... |
In this paper we present uniform weighted inequalities for the Hankel transform transplantation operator. A weighted vector-valued inequality is also obtained. |
In this paper we prove weighted mixed norm estimates for Riesz transforms associated to Hermite and special Hermite operators. The estimates are shown to be equivalent to vectorvalued esimates for a sequence of operators defined in terms of Laguerre functions of different type. |
Using factorisation and Arov-Krein inequality results, we derive important inequalities (in terms of $S$-nodes) in interpolation problems. |
In this article, we derive multiple polylogarithms from multiple zeta values by using a recursive Riemann-Hilbert problem of additive type. Furthermore we show that this Riemann-Hilbert problem is regarded as an inverse problem for the connection problem of the KZ equation of one variable, so that the fundamental solu... |
We prove genuinely sharp estimates for the Jacobi heat kernels introduced in the context of the multidimensional cone $\mathbb{V}^{d+1}$ and its surface $\mathbb{V}^{d+1}_0$. To do so, we combine the theory of Jacobi polynomials on the cone explored by Xu with the recent techniques by Nowak, Sjögren, and Szarek, devel... |
In this paper, we study joint functional calculus for commuting $n$-tuple of Ritt operators. We provide an equivalent characterisation of boundedness for joint functional calculus for Ritt operators on $L^p$-spaces, $1< p<\infty$. |
The operators $\Lambda_m$ ($m\in\mathbb{N}\cup \{0\}$) arise when one studies the action of the Beurling-Ahlfors transform on certain radial function subspaces. It is known that the weak-type $(1,1)$ constant of $\Lambda_0$ is equal to $1/\ln(2)\approx 1.44$. |
While teaching a course on integral equations, I noticed that a straightforward combination of Neumann series and Fourier series for the resolvent (or the solution) of an integral equation has good approximation qualities. This short article presents and investigates this combination of approximating series. |
We obtain an improved Kakeya maximal function estimate in $\mathbb{R}^4$ using a new geometric argument called the planebrush. A planebrush is a higher dimensional analogue of Wolff's hairbrush, which gives effective control on the size of Besicovitch sets when the lines through a typical point concentrate into a ... |
TThe main aim of this paper is to prove that there exist a martingale $f\in H_{1/2}$ such that Fejér means of Vilenkin-Fourier series of the martingale $f$ is not uniformly bounded in the space $L_{1/2}. $ |
We generalize to the setting of 1-sided chord-arc domains, that is, to domains satisfying the interior Corkscrew and Harnack Chain conditions (these are respectively scale-invariant/quantitative versions of the openness and path-connectedness) and which have an Ahlfors regular boundary, a result of Kenig-Kirchheim-Piph... |
One solution to a relatively recent American Mathematical Monthly problem [6], requesting the evaluation of a real definite integral, could be couched in terms of a contour integral which vanishes {\textit{a priori. }} While the required real integral emerged on setting to zero the real part of the contour quadrature, ... |
In this paper, we represent a continued fraction expression of Mathieu series by a continued fraction formula of Ramanujan. As application, we obtain some new bounds for Mathieu series. |
In an article that appeared in Journal of Approximation Theory, Neuschel investigated the asymptotic expansion of certain hypergeometric polynomials of type ${}_3 F_1$ inside and outside a closed curve. We supplement this result by studying a subfamily of those polynomials on a part of the closed curve. |
It is known that the Funk transform (the Funk-Radon transform) is invertible in the class of even (symmetric) continuous functions defined on the unit 2-sphere S^2. In this article, for the reconstruction of f from C(S^2) (can be non-even), an additional condition (to reconstruct an odd function) is found, and the inj... |
This note is developing, and we will include many additions in the near future. Our purpose here is to highlight that there is plenty of space for a topological development of the Fermat Real Line. |
Let $E\subset \mathbb{R}^{n+1}$, $n\ge 2$, be an Ahlfors-David regular set of dimension $n$. We show that the weak-$A_\infty$ property of harmonic measure, for the open set $\Omega:= \mathbb{R}^{n+1}\setminus E$, implies uniform rectifiability of $E$. |
We consider multiple Geronimus transformations and show that they lead to discrete (non-diagonal) Sobolev type inner products. Moreover, it is shown that every discrete Sobolev inner product can be obtained as a multiple Geronimus transformation. |
In this article, we generalize a theorem of Victor L. Shapiro concerning nontangential convergence of the Poisson integral of a $L^p$-function. We introduce the notion of $\sigma$-points of a locally finite measure and consider a wide class of convolution kernels. |
We prove that for any $q\in (0,1)$, all complex conjugate pairs of zeros of the partial theta function $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ with non-negative real part belong to the half-annulus $\{$Re$(x)\geq 0,~1<|x|<5\}$, where the outer radius cannot be replaced by a number smaller than $e^{\pi... |
We prove that $$\max_{t \in [-\pi,\pi]}{|Q(t)|} \leq T_{2n}(\sec(s/4)) = \frac 12 ((\sec(s/4) + \tan(s/4))^{2n} + (\sec(s/4) - \tan(s/4))^{2n})$$ for every even trigonometric polynomial $Q$ of degree at most $n$ with complex coefficients satisfying $$m(\{t \in [-\pi,\pi]: |Q(t)| \leq 1\}) \geq 2\pi-s\,, \qquad s \in (0... |
In this article we continue the work from <a href="https://arxiv.org/abs/0902.0621" data-arxiv-id="0902.0621" class="link-https">arXiv:0902.0621</a>. In that article Eric Rains and the present author considered the limits of the elliptic beta integral as p->0 while the parameters t_r have a p-dependence of the form... |
In this paper, we show that the 2-sphere does not exhibit symmetry of $L^p$ norms of eigenfunctions of the Laplacian for $p\geq 6$. In other words, there exists a sequence of spherical eigenfunctions $\psi_n$, with eigenvalues $\lambda_n\to\infty$ as $n\to\infty$, such that the ratio of the $L^p$ norms of the positive... |
In this paper, necessary and sufficient conditions on terms of monotone Fourier coefficients for a function to belong to a Nikol'ski\uı--Besov type class are given. |
By a nondegenerate $k$-parameterized family $K$ of periodic solutions we understand the situation when the geometric multiplicity of the multiplier +1 of the linearized on $K$ system equals to $k. $ Bifurcation of asymptotically stable periodic solutions from $K$ is well studied in the literature and different conditio... |
The recently introduced concept of $\mathcal{D}$-variation unifies previous concepts of variation of multivariate functions. In this paper, we give an affirmative answer to the open question from Pausinger \& Svane (J. Complexity, 2014) whether every function of bounded Hardy--Krause variation is Borel measurable ... |
A nonnegative real function f is bell-shaped if it converges to zero at plus and minus infinity and the nth derivative of f changes sign n times for every n = 0, 1, 2, ... Similarly, a two-sided nonnegative sequence a(k) is bell-shaped if it converges to zero at plus and minus infinity and the nth iterated difference ... |
A simple shortcut to proving sharp weighted estimates for the Martingale Transform and for the dyadic shift of order 1 (and so for the Hilbert transform) is presented. It is a unified proof for these both transforms. |
We prove some new results and unify the proofs of old ones involving complete monotonicity of expressions involving gamma and $q$-gamma functions, $0 < q < 1$. Each of these results implies the infinite divisibility of a related probability measure. |
Let Pd,n denote the space of all real polynomials of degree at most d on R^n. We prove a new estimate for the logarithmic measure of the sublevel set of a polynomial P in Pd,1. |
A Lagrangian method is introduced recently for deriving indefinite integrals of special functions that satisfy homogeneous (nonhomogeneous) second-order linear differential equations. This paper extends this method to include indefinite Jackson $q$-integrals of special functions satisfying homogeneous (nonhomogeneous)... |
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