Geodesic
Gauss's binomial formula and additive property of exponential functions on T (q,h)
Filomat, Tome 35 (2021) no. 11, p. 3855

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DOI

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,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
DOI : 10.2298/FIL2111855S
Classification : 39A06, 39A13, 39A14, 34NA05
Keywords: (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 
Burcu Silindir; Ahmet Yantir. Gauss's binomial formula and additive property of exponential functions on T (q,h). Filomat, Tome 35 (2021) no. 11, p. 3855 . doi: 10.2298/FIL2111855S
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     author = {Burcu Silindir and Ahmet Yantir},
     title = {Gauss's binomial formula and additive property of exponential functions on {T} (q,h)},
     journal = {Filomat},
     pages = {3855 },
     year = {2021},
     volume = {35},
     number = {11},
     doi = {10.2298/FIL2111855S},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.2298/FIL2111855S/}
}
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