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\(q\)-extension of the Euler formula and trigonometric functions. (English) Zbl 1188.33001

Replacing the factorial \(n!\) by the \(q\)-factorial \[ [n]_q!=\prod_{k=1}^n\frac{1-q^k}{1-q} \] the author defines the \(q\)-exponential function \[ e_q(z)=\sum_{n=0}^\infty\frac{z^n}{[n]_q!} \] and the related \(q\)-trigonometric functions \(\cos_q(z)\) and \(\sin_q(z)\) as the real and imaginary parts of \(e_q(iz)\). Then he derives \(q\)-analogues of classical formulae satisfied by these functions.

MSC:

33B10 Exponential and trigonometric functions
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
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References:

[1] Kim, T., Power Series and Asymptotic Series Associated with the q-Analog of the Two-Variable p-Adic L-Function, Russ. J. Math. Phys., 12, 2, 186-196 (2005) · Zbl 1190.11049
[2] Kim, T., An Invariant p-Adic Integral Associated with Daehee Numbers, Integral Transforms Spec. Funct., 13, 65-69 (2002) · Zbl 1016.11008
[3] Kim, T.; Rim, S. H., A Note on the q-Integral and q-Series, Adv. Stud. Contemp. Math., 2, 37-45 (2000) · Zbl 0976.05008
[4] Koornwinder, T. H., Special Functions and q-Commuting Variables, Fields Inst. Commun., 14, 131-166 (1997) · Zbl 0882.33014
[5] E. Kreyszig, Advanced Engineering Mathematics, 8th ed. (John Wiley & Sons, 1999). · Zbl 0103.27803
[6] Schork, M., Wards ‘Calculus of Sequences’ q-Calculus and the Limit q → 1, Adv. Stud. Contemp. Math., 13, 131-141 (2006) · Zbl 1111.05010
[7] Simsek, Y., q-Dedekind Type Sums Related to q-Zeta Function and Basic L-Series, J. Math. Anal. Appl., 318, 333-351 (2006) · Zbl 1149.11054
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