The main purpose of this paper is to introduce and investigate a class of -Bernoulli, -Euler, and -Genocchi polynomials. The -analogues of well-known formulas are derived. In addition, the -analogue of the Srivastava-Pintér theorem is obtained. Some new identities, involving -polynomials, are proved.

Our study is based on the modification of a well-known predator-prey equation, or the Lotka–Volterra competition model. That is, a system of differential equations was established for the population of healthy and cancerous cells within the tumor tissue of a patient struggling with cancer. Besides, fractional differentiation remedies the situation by obtaining a meticulous model with more flexible parameters. Furthermore, a specific type of non-Newtonian calculus, bi-geometric calculus, can describe the model in terms of proportions and implies the alternative aspect of a dynamic system. Moreover, fractional operators in bi-geometric calculus are formulated in terms of Hadamard fractional operators. In this article, the development of fractional operators in non-Newtonian calculus was investigated. The model was extended in these criteria, and the existence and uniqueness of the model were considered and guaranteed in the first step by applying the Arzelà–Ascoli. The bi-geometric analogue of the numerical method provided a suitable tool to solve the model approximately. In the end, the visual graphs were obtained by using the MATLAB program.

We study the q-analogue of Euler-Maclaurin formula and by introducing a new q-operator we drive to this form. Moreover, approximation properties of q-Bernoulli polynomials are discussed. We estimate the suitable functions as a combination of truncated series of q-Bernoulli polynomials and the error is calculated. This paper can be helpful in two different branches: first we solve the differential equations by estimating functions and second we may apply these techniques for operator theory.

The Non-Newtonian calculus is inspired by the homeomorphism from the real line to the open interval that we denote it by α(x). This function modifies the preliminary algebraic operators and the new fractional differintegration formula will be found. On the other hand, the Ψ-fractional calculus was introduced in the aid of generalized Leibniz formula for production of two differentiable functions which has been driven to the similar fractional operators. In this literature the relationship between these two operators are investigated and new interpretation of Ψ-fractional calculus is studied.

In this paper, we study on the new class of q-fractional integral operator. In the aid of iterated Cauchy integral approach to fractional integral operator, we applied tp f(t) in these integrals and a new class of q-fractional integral operator with parameter p, is introduced. Recently, the q-analogue of fractional differential integral operator is studied and all of the operators defined in these studies are q-analogue of Riemann fractional differential operator. We show that our new class of operator generalize all the operators in use, and additionally, it can cover the q-analogue of Hadamard fractional differential operator, as well. Some properties of this operator are investigated.

In this article, new class of q-fractional integral operator is introduced. In the aid of this definition, related derivative is investigated and inverse properties are studied. New operator based on q-fractional integral operator is created and a class of (p, q)-absolutely continous function is classified.