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q-exponential function

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1: 17.17 Physical Applications
β–ΊSee Kassel (1995). …
2: 17.3 q -Elementary and q -Special Functions
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q -Exponential Functions
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17.3.1 e q ⁑ ( x ) = n = 0 ( 1 q ) n ⁒ x n ( q ; q ) n = 1 ( ( 1 q ) ⁒ x ; q ) ,
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17.3.2 E q ⁑ ( x ) = n = 0 ( 1 q ) n ⁒ q ( n 2 ) ⁒ x n ( q ; q ) n = ( ( 1 q ) ⁒ x ; q ) .
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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 .
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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
β–ΊFor computation of the q -exponential function see Gabutti and Allasia (2008). …
4: Bibliography O
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  • 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
    β–ΊThroughout this section q and ΞΆ are defined as in §22.2. β–ΊIf q ⁒ exp ⁑ ( 2 ⁒ | ⁑ ΞΆ | ) < 1 , then … β–ΊNext, if q ⁒ exp ⁑ ( | ⁑ ΞΆ | ) < 1 , then … β–ΊNext, with E ⁑ = E ⁑ ( k ) denoting the complete elliptic integral of the second kind (§19.2(ii)) and q ⁒ exp ⁑ ( 2 ⁒ | ⁑ ΞΆ | ) < 1 , … β–Ί
    6: 28.14 Fourier Series
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    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
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    §20.11(ii) Ramanujan’s Theta Function and q -Series
    β–ΊWith the substitutions a = q ⁒ e 2 ⁒ i ⁒ z , b = q ⁒ e 2 ⁒ i ⁒ z , with q = e i ⁒ Ο€ ⁒ Ο„ , we have … β–ΊIn 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). … β–ΊAs 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). … β–ΊFor m = 1 , 2 , 3 , 4 , n = 1 , 2 , 3 , 4 , and m n , define twelve combined theta functions Ο† m , n ⁑ ( z , q ) by …
    8: 18.28 Askey–Wilson Class
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    18.28.3 2 ⁒ Ο€ ⁒ sin ⁑ ΞΈ ⁒ w ⁑ ( cos ⁑ ΞΈ ) = | ( e 2 ⁒ i ⁒ ΞΈ ; q ) ( a ⁒ e i ⁒ ΞΈ , b ⁒ e i ⁒ ΞΈ , c ⁒ e i ⁒ ΞΈ , d ⁒ e i ⁒ ΞΈ ; q ) | 2 ,
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    18.28.8 1 2 ⁒ Ο€ ⁒ 0 Ο€ Q n ⁑ ( cos ⁑ ΞΈ ; a , b | q ) ⁒ Q m ⁑ ( cos ⁑ ΞΈ ; a , b | q ) ⁒ | ( e 2 ⁒ i ⁒ ΞΈ ; q ) ( a ⁒ e i ⁒ ΞΈ , b ⁒ e i ⁒ ΞΈ ; q ) | 2 ⁒ d ΞΈ = Ξ΄ n , m ( q n + 1 , a ⁒ b ⁒ q n ; q ) , a , b ℝ or a = b ¯ ; | a ⁒ b | < 1 ; | a | , | b | 1 .
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    18.28.15 1 2 ⁒ Ο€ ⁒ 0 Ο€ C n ⁑ ( cos ⁑ ΞΈ ; Ξ² | q ) ⁒ C m ⁑ ( cos ⁑ ΞΈ ; Ξ² | q ) ⁒ | ( e 2 ⁒ i ⁒ ΞΈ ; q ) ( Ξ² ⁒ e 2 ⁒ i ⁒ ΞΈ ; q ) | 2 ⁒ d ΞΈ = ( Ξ² , Ξ² ⁒ q ; q ) ( Ξ² 2 , q ; q ) ⁒ ( 1 Ξ² ) ⁒ ( Ξ² 2 ; q ) n ( 1 Ξ² ⁒ q n ) ⁒ ( q ; q ) n ⁒ Ξ΄ n , m , 1 < Ξ² < 1 .
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    18.28.17 1 2 ⁒ Ο€ ⁒ 0 Ο€ H n ⁑ ( cos ⁑ ΞΈ | q ) ⁒ H m ⁑ ( cos ⁑ ΞΈ | q ) ⁒ | ( e 2 ⁒ i ⁒ ΞΈ ; q ) | 2 ⁒ d ΞΈ = Ξ΄ n , m ( q n + 1 ; q ) .
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    18.28.18 h n ⁑ ( sinh ⁑ t | q ) = β„“ = 0 n q 1 2 ⁒ β„“ ⁒ ( β„“ + 1 ) ⁒ ( q n ; q ) β„“ ( q ; q ) β„“ ⁒ e ( n 2 ⁒ β„“ ) ⁒ t = e n ⁒ t ⁒ Ο• 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
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    18.27.20 0 S n ⁑ ( q 1 2 ⁒ x ; q ) ⁒ S m ⁑ ( q 1 2 ⁒ x ; q ) ⁒ exp ⁑ ( ( ln ⁑ x ) 2 2 ⁒ ln ⁑ ( q 1 ) ) ⁒ d x = 2 ⁒ Ο€ ⁒ q 1 ⁒ ln ⁑ ( q 1 ) q n ⁒ ( q ; q ) n ⁒ Ξ΄ n , m .