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In this article, we construct a new type of degenerate q-sigmoid (DQS) polynomial for sigmoid functions containing quantum numbers and find several difference equations related to it. We check how each point moves by iteratively synthesizing a quartic degenerate q-sigmoid (DQS) polynomial that appears differently depending on q in the space of a complex structure. We also construct Julia sets associated with quartic DQS polynomials and find their features. Based on this, we make some conjectures. [ABSTRACT FROM AUTHOR]

In this paper, we generate new degenerate quantum Euler polynomials (DQE polynomials), which are related to both degenerate Euler polynomials and q-Euler polynomials. We obtain several (q , h) -differential equations for DQE polynomials and find some relations of q-differential and h-differential equations. By varying the values of q , η , and h, we observe the values of DQE numbers and approximate roots of DQE polynomials to obtain some properties and conjectures. [ABSTRACT FROM AUTHOR]

In this paper, we construct degenerate q-tangent numbers and polynomials and determine their related properties. Based on these numbers and polynomials, we also confirm that the structure of the approximate root changes according to changes in q and h. We find differential equations that have degenerate q-tangent polynomials as solutions and also find differential equations that have other polynomials as coefficients, confirming the relationships among these. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to organize various types of higher order q-differential equations that are connected to q-sigmoid polynomials and obtain certain properties regarding their solutions. Using the properties of q-sigmoid polynomials, we show the symmetric properties of q-differential equations of higher order. Moreover, we derive special properties for the approximate roots of q-sigmoid polynomials which are solutions of higher order q-differential equations. [ABSTRACT FROM AUTHOR]

This paper intends to define degenerate q-Hermite polynomials, namely degenerate q-Hermite polynomials by means of generating function. Some significant properties of degenerate q-Hermite polynomials such as recurrence relations, explicit identities and differential equations are established. Many mathematicians have been studying the differential equations arising from the generating functions of special numbers and polynomials. Based on the results so far, we find the differential equations for the degenerate q-Hermite polynomials. We also provide some identities for the degenerate q-Hermite polynomials using the coefficients of this differential equation. Finally, we use a computer to view the location of the zeros in degenerate q-Hermite equations. Numerical experiments have confirmed that the roots of the degenerate q-Hermit equations are not symmetric with respect to the imaginary axis. [ABSTRACT FROM AUTHOR]

One finds several q-differential equations of a higher order for q-Euler polynomials and q-Genocchi polynomials. Additionally, we have a few q-differential equations of a higher order, which are mixed with q-Euler numbers and q-Genocchi polynomials. Moreover, we investigate some symmetric q-differential equations of a higher order by applying symmetric properties of q-Euler polynomials and q-Genocchi polynomials. [ABSTRACT FROM AUTHOR]

The sigmoid polynomials combining q -numbers and various properties of their polynomials are thoroughly presented in this paper. Based on several properties of q -numbers, we derive some identities of q -sigmoid polynomials. Also, from explicit polynomials, we can observe structures of the zeros of the sigmoid polynomials which are related to q -numbers. [ABSTRACT FROM AUTHOR]

In this paper, we define (p , q) -cosine and sine sigmoid polynomials. Based on this, the properties of each polynomial, and the structure and assumptions of its roots, can be identified. Properties can also be determined by the changes in p and q. [ABSTRACT FROM AUTHOR]

In this paper, we introduce (p , q) -cosine Euler polynomials. From these polynomials, we find several properties and identities. Moreover, we find the circle equations of approximate roots for (p , q) -cosine Euler polynomials by using a computer. [ABSTRACT FROM AUTHOR]

In this paper, we introduce q-cosine and q-sine Euler polynomials and determine identities for these polynomials. From these polynomials, we obtain some special properties using a power series of q-trigonometric functions, properties of q-exponential functions, and q-analogues of the binomial theorem. We investigate the approximate roots of q-cosine Euler polynomials that help us understand these polynomials. Moreover, we display the approximate roots movements of q-cosine Euler polynomials in a complex plane using the Newton method. [ABSTRACT FROM AUTHOR]

This paper constructs and introduces (p , q) -cosine and (p , q) -sine Bernoulli polynomials using (p , q) -analogues of (x + a) n . Based on these polynomials, we discover basic properties and identities. Moreover, we determine special properties using (p , q) -trigonometric functions and verify various symmetric properties. Finally, we check the symmetric structure of the approximate roots based on symmetric polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we define cosine Bernoulli polynomials and sine Bernoulli polynomials related to the q-number. Furthermore, we intend to find the properties of these polynomials and check the structure of the roots. Through numerical experimentation, we look for various assumptions about the polynomials above. [ABSTRACT FROM AUTHOR]