Gauss's Binomial Formula and Additive Property of Exponential Functions on T-(q,T-h)

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Silindir Yantır B., Yantır A.

FILOMAT, cilt.35, sa.11, ss.3855-3877, 2021 (SCI-Expanded) identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 35 Sayı: 11
  • Basım Tarihi: 2021
  • Doi Numarası: 10.2298/fil2111855s
  • Dergi Adı: FILOMAT
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, zbMATH
  • Sayfa Sayıları: ss.3855-3877
  • Anahtar Kelimeler: (q,h)-Gauss's binomial formula, (q, h)-integral, (q, h)-analytic functions, additive property of (q, h)-exponential functions, (q, h)-trigonometric functions, (q, h)-diffusion equation, (q, h)-Burger's equation
  • Dokuz Eylül Üniversitesi Adresli: Evet

Özet

In this article, we focus our attention on (q, h)-Gauss's binomial formula from which we discover the additive property of (q ,h)-exponential functions. We state the (q, h)-analogue of Gauss's binomial formula in terms of proper polynomials on T-(q,T-h) which own essential properties similar to ordinary polynomials. We present (q, h)-Taylor series and analyze the conditions for its convergence. We introduce a new (q, h)-analytic exponential function which admits the additive property. As consequences, we study (q, h)-hyperbolic functions, (q, h)-trigonometric functions and their significant properties such as (q, h)-Pythagorean Theorem and double-angle formulas. Finally, we illustrate our results by a first order (q, h)-difference equation, (q, h)-analogues of dynamic diffusion equation and Burger's equation. Introducing (q, h)-Hopf-Cole transformation, we obtain (q, h)-shock soliton solutions of Burger's equation.