On the Exponential and Trigonometric $$q,\omega $$-Special Functions

Ernst, Thomas

doi:10.1007/978-3-030-41850-2_27

Cited by 2 publications

(4 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Of course we have R q ⊂ R q , although these number systems are quite different. Several of the results in the last section on q-trigonometric and hyperbolic functions were previously published in more general q, ω-form in our paper [11]. Since the graphs of Tan q (x) and Cot q (x) closely resemble the graphs of the corresponding q, ω-functions, we have skipped these two graphs.…”

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

Three algebraic number systems based on the q-addition with applications

Ernst

2022

View full text Add to dashboard Cite

In the spirit of our earlier articles on q-ω special functions, the purpose of this article is to present many new q-number systems, which are based on the q-addition, which was introduced in our previous articles and books. First, we repeat the concept biring, in order to prepare for the introduction of the q-integers, which extend the q-natural numbers from our previous book. We formally introduce a q-logarithm for the q-exponential function for later use. In order to find q-analogues of the corresponding formulas for the generating functions and q-trigonometric functions, we also introduce q-rational numbers. Then the so-called q-real numbers R⊕ q , with a norm, a q-deformed real line, and with three inequalities, are defined. The purpose of the more general q-real numbers Rq is to allow the other q-addition too. The closely related JHC q-real numbers R q have applications to several q-Euler integrals. This brings us to a vector version of the q-binomial theorem from a previous paper, which is associated with a special case of the q-Lauricella function. New q-trigonometric function formulas are given to show the application of this umbral calculus. Then, some equalities between q-trigonometric zeros and extreme values are proved. Finally, formulas and graphs for q-hyperbolic functions are shown.

show abstract

Section: Discussionmentioning

confidence: 99%

“…There are also extra multiple products on the right hand side. As soon as we have ⊕ q instead of ⊕ in formula (11), the result becomes smaller as in the proof of (10). We shall give one example of a q-real number, where the norm in ( 8) is infinite in formula (5).…”

Section: The Q-real Numbers R ⊕Q and R Qmentioning

confidence: 99%

Three algebraic number systems based on the q-addition with applications

Ernst

2022

View full text Add to dashboard Cite

show abstract

“…The following names will be used for the ensuing q, ω-trigonometric and hyperbolic functions [8]. The two following formulas correspond to the formula Dx n = nx n−1 : [11, 2.5], [21, (17)…”

Section: Thomas Ernstmentioning

confidence: 99%

“…The convergence region in ω will always be a small interval above 0, depending on q. The subtle properties of absolute maximum for the two q, ω-additions are exemplified in [8].…”

Section: Introductionmentioning

confidence: 99%

On Three General q, w- Apostol Polynomials and their Connection to q, w- Power Sums

Ernst¹

2019

Advances in Dynamical Systems and Applications

Self Cite

View full text Add to dashboard Cite

The purpose of this article is to continue our study of q, ω-special functions, in particular three general q, ω-Apostol polynomials, which contain an extra parameter λ. We find complementary argument theorems, multiplication formulas and recurrence relations for the corresponding numbers. Furthermore, an explicit formula for the multiple alternating q, ω-power sum is found. These formulas are then specialized to q, ω-Apostol-Bernoulli and Euler polynomials.

show abstract

On the Exponential and Trigonometric $$q,\omega $$-Special Functions

Cited by 2 publications

References 4 publications

Three algebraic number systems based on the q-addition with applications

Three algebraic number systems based on the q-addition with applications

On Three General q, w- Apostol Polynomials and their Connection to q, w- Power Sums

Contact Info

Product

Resources

About