Related papers: Summable solutions of some partial differential eq…
A generalization of the classical Lipschitz summation formula is proposed. It involves new polylogarithmic rational functions constructed via the Fourier expansion of certain sequences of Bernoulli--type polynomials. Related families of…
We consider a family of linear singularly perturbed Cauchy problems which combines partial differential operators and linear fractional transforms. We construct a collection of holomorphic solutions on a full covering by sectors of a…
The objective of this paper is to investigate the existence and the forms of the pair of finite order entire and meromorphic solutions of some certain systems of Fermat-type partial differential-difference equations of several complex…
We introduce a general multisummability theory of formal power series in Carleman ultraholomorphic classes. The finitely many levels of summation are determined by pairwise comparable, nonequivalent weight sequences admitting nonzero…
The method of separation of variables can be used to solve many separable linear partial differential equations (LPDEs). Moreover, variable separation solutions usually are some trigonometric series. In the paper, base on some ideas of this…
It is well known that the real and imaginary parts of any holomorphic function are harmonic functions of two variables. In this paper we generalize this property to finite-dimensional commutative algebras. We prove that if some basis of a…
This paper is devoted to strictly hyperbolic systems and equations with non-smooth coefficients. Below a certain level of smoothness, distributional solutions may fail to exist. We construct generalised solutions in the Colombeau algebra of…
The paper presents a new formula for the fractional integration, which generalizes the Riemann-Liouville and Hadamard fractional integrals into a single form, which when a parameter fixed at different values, produces the above integrals as…
One-parameter generalizations of the logarithmic and exponential functions have been obtained as well as algebraic operators to retrieve extensivity. Analytical expressions for the successive applications of the sum or product operators on…
By means of the derivative operator and Whipple-type $_3F_2$-series identities, two families of summation formulae involving generalized harmonic numbers are established.
In this paper are examined general classes of linear and non-linear analytical systems of partial differential equations. Indeed the integrability conditions are found and if they are satisfied, the solutions are given as functional series…
We present an extension of a previously developed method employing the formalism of the fractional derivatives to solve new classes of integral equations. This method uses different forms of integral operators that generalizes the…
Using increasing sequences of real numbers, we generalize the idea of formal moment differentiation first introduced by W. Balser and M. Yoshino. Slight departure from the concept of Gevrey sequences enables us to include a wide variety of…
The paper proposes a vector generalization of the basic concepts of the theory of complex variable: the concept of modulus and argument of complex number. The author introduces some generalizations of the notion of holomorphic functions and…
The concept of weighted $\beta\gamma$ - summability of order $\theta$ in case of fuzzy functions is introduced and classified into ordinary and absolute sense. Several inclusion relations among the sets are investigated. Also we have found…
This paper develops one of the methods for study of nonlinear Partial Differential equations. We generalize Sato equation and represent the algorithm for construction of some classes of nonlinear Partial Differential Equations (PDE)…
Harmonic sums and their generalizations are extremely useful in the evaluation of higher-order perturbative corrections in quantum field theory. Of particular interest have been the so-called nested sums,where the harmonic sums and their…
Several algebro-geometric properties of commutative rings of partial differential operators as well as several geometric constructions are investigated. In particular, we show how to associate a geometric data by a commutative ring of…
By applying the derivative operator to the corresponding hypergeometric form of a $q$-series transformation due to Andrews [1,Theorem 4], we establish a general harmonic number identity. As the special cases of it, several interesting…
We give a review of modern approaches to constructing formal solutions to integrable hierarchies of mathematical physics, whose coefficients are answers to various enumerative problems. The relationship between these approaches and…