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q-exponential function

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1: 17.17 Physical Applications
See Kassel (1995). …
2: 17.3 q -Elementary and q -Special Functions
q -Exponential Functions
17.3.1 e q ( x ) = n = 0 ( 1 - q ) n x n ( q ; q ) n = 1 ( ( 1 - q ) x ; q ) ,
17.3.2 E q ( x ) = n = 0 ( 1 - q ) n q ( n 2 ) x n ( q ; q ) n = ( - ( 1 - q ) x ; q ) .
17.3.4 Sin q ( x ) = 1 2 i ( E q ( i x ) - E q ( - i x ) ) = n = 0 ( 1 - q ) 2 n + 1 q n ( 2 n + 1 ) ( - 1 ) n x 2 n + 1 ( q ; q ) 2 n + 1 .
17.3.6 Cos q ( x ) = 1 2 ( E q ( i x ) + E q ( - i x ) ) = n = 0 ( 1 - q ) 2 n q n ( 2 n - 1 ) ( - 1 ) n x 2 n ( q ; q ) 2 n .
3: 17.18 Methods of Computation
For computation of the q -exponential function see Gabutti and Allasia (2008). …
4: Bibliography O
  • A. B. Olde Daalhuis (1994) Asymptotic expansions for q -gamma, q -exponential, and q -Bessel functions. J. Math. Anal. Appl. 186 (3), pp. 896–913.
  • 5: 22.11 Fourier and Hyperbolic Series
    Throughout this section q and ζ are defined as in §22.2. If q exp ( 2 | ζ | ) < 1 , then … Next, if q exp ( | ζ | ) < 1 , then … Next, with E = E ( k ) denoting the complete elliptic integral of the second kind (§19.2(ii)) and q exp ( 2 | ζ | ) < 1 , …
    6: 28.14 Fourier Series
    28.14.1 me ν ( z , q ) = m = - c 2 m ν ( q ) e i ( ν + 2 m ) z ,
    7: 20.11 Generalizations and Analogs
    §20.11(ii) Ramanujan’s Theta Function and q -Series
    With the substitutions a = q e 2 i z , b = q e - 2 i z , with q = e i π τ , we have … In the case z = 0 identities for theta functions become identities in the complex variable q , with | q | < 1 , that involve rational functions, power series, and continued fractions; see Adiga et al. (1985), McKean and Moll (1999, pp. 156–158), and Andrews et al. (1988, §10.7). … As in §20.11(ii), the modulus k of elliptic integrals (§19.2(ii)), Jacobian elliptic functions22.2), and Weierstrass elliptic functions23.6(ii)) can be expanded in q -series via (20.9.1). … For m = 1 , 2 , 3 , 4 , n = 1 , 2 , 3 , 4 , and m n , define twelve combined theta functions φ m , n ( z , q ) by …
    8: 18.28 Askey–Wilson Class
    18.28.3 2 π sin θ w ( cos θ ) = | ( e 2 i θ ; q ) ( a e i θ , b e i θ , c e i θ , d e i θ ; q ) | 2 ,
    18.28.8 1 2 π 0 π Q n ( cos θ ; a , b | q ) Q m ( cos θ ; a , b | q ) | ( e 2 i θ ; q ) ( a e i θ , b e i θ ; q ) | 2 d θ = δ n , m ( q n + 1 , a b q n ; q ) , a , b or a = b ¯ ; | a b | < 1 ; | a | , | b | 1 .
    18.28.15 1 2 π 0 π C n ( cos θ ; β | q ) C m ( cos θ ; β | q ) | ( e 2 i θ ; q ) ( β e 2 i θ ; q ) | 2 d θ = ( β , β q ; q ) ( β 2 , q ; q ) ( 1 - β ) ( β 2 ; q ) n ( 1 - β q n ) ( q ; q ) n δ n , m , - 1 < β < 1 .
    18.28.18 h n ( sinh t | q ) = = 0 n q 1 2 ( + 1 ) ( q - n ; q ) ( q ; q ) e ( n - 2 ) t = e n t ϕ 1 1 ( q - n 0 ; q , - q e - 2 t ) = i - n H n ( i sinh t | q - 1 ) .
    9: 18.27 q -Hahn Class