An Extension of Caputo Fractional Derivative Operator by Use of Wiman’s Function

The main aim of this work is to study an extension of the Caputo fractional derivative operator by use of the two-parameter Mittag–Leffler function given by Wiman. We have studied some generating relations, Mellin transforms and other relationships with extended hypergeometric functions in order to derive this extended operator. Due to symmetry in the family of special functions, it is easy to study their various properties with the extended fractional derivative operators.

The incomplete exponential pRp (α,β;z) function with applications

In this paper, we are motivated to establish the generalization of a R q p ⁢ ( α , β ; z ) {{}_{p}R_{q}(\alpha,\beta;z)} function in terms of incomplete exponential functions and obtain some properties of an incomplete exponential R q p ⁢ ( α , β ; z ) {{}_{p}R_{q}(\alpha,\beta;z)} function. Further we give generating relations for incomplete exponential R q p ⁢ ( α , β ; z ) {{}_{p}R_{q}(\alpha,\beta;z)} function. Some applications related to ground water pumping (hydrology) modelling and probability theory are also discussed.

Generating functions involving the incomplete H-functions

In this article, for the incomplete H-functions, we obtain a set of new generating functions. The bilateral along with linear generating relations are derived for the incomplete H-functions. Many of the generating functions readily accessible in the literature are often deemed as implementations of the main findings. All the derived findings are of a natural type and can produce a variety of new results in generating function theory.

Artin–Schelter regular algebras of dimension five with three generators

In this paper, we investigate Artin–Schelter regular algebras of dimension [Formula: see text] with three generators in degree [Formula: see text] under the hypothesis that [Formula: see text], in which the degree types of the relations for the number of the generating relations less than five can be determined. We prove that the only possible degree type of three generating relations is [Formula: see text] and the only possible degree type of four generating relations is [Formula: see text].

Generating Functions for Some Hypergeometric Functions of Four Variables via Laplace Integral Representations

Generating functions plays an essential role in the investigation of several useful properties of the sequences which they generate. In this paper, we establish certain generating relations, involving some quadruple hypergeometric functions introduced by Bin-Saad and Younis. Some interesting special cases of our main results are also considered.

Word-formation motivation as a means of lexicalizing derivative words in the classroom in Russian as a foreign language

The article examines the place of the word-formation component of the communicative competence of foreign students in the process of their learning Russian as a foreign language. The word-formation component, along with other linguistic components, participates in the formation of the communicative competence of foreign students, however, the problem of Russian word formation remains insufficiently developed in the linguodidactic plan. The author expresses his opinion that the increase in the vocabulary of foreign students should be accompanied by the systematization of lexical units, taking into account their derivational connections and relations. Therefore, when lexicalization, one should use not the selection of synonyms or the descriptive method, which leads to the mechanical memorization of the meaning, but by means of correlation with the same root words, to activate in the minds of foreign students the living connections of the lexical units of the Russian language. This applies not only to derivatives of words associated with generating relations of full direct motivation, but also to those derivatives where these connections are not explicitly expressed. Such an interpretation through a productive word not only makes the meaning of the new lexeme understandable, facilitates its memorization, but also creates a single picture of lexical and word-formation relationships within the language, forms a systemic understanding of the language among students. The skills of correlating a derivative word with its derivative, obtained in the study of Russian as a foreign language, make it possible to understand the meanings of unfamiliar words. The speed of memorizing the meanings of derived words in this case directly depends on the understanding of the word-formation structure of the word. This technique will greatly facilitate the development of the vocabulary and will contribute to the formation of a holistic view of the language as a system among foreign students.

A sequence involving an extended Struve function via a differential operator

Various extensions of the Struve function have been presented and investigated. Here we aim to introduce an extended Struve function involving the $\mathtt{k}$-gamma function. Then, by using a known differential operator, we introduce a sequence of functions associated with the above introduced extended Struve function and investigate its properties such as generating relations and a finite summation formula. The results presented here, being very general, are also pointed out to yield a number of relatively simple identities.

Correction: Gürel Yılmaz, Ö., et al. On Some Formulas for the k-Analogue of Appell Functions and Generating Relations via k-Fractional Derivative. Fractal Fract. 2020, 4, 48

The authors wish to make the following corrections to this paper [...]

Relations between modular invariants of a vector and a covector in dimension two

We exhibit a set of generating relations for the modular invariant ring of a vector and a covector for the two-dimensional general linear group over a finite field.