/* -*- gp-script -*- */ \\% q-exponential and q-trigonometric series of the first kind. \\ Author: Joerg Arndt \\ License: GPL version 3 or later \\ online at http://www.jjj.de/pari/ \\ version: 2014-October-16 (18:30) \\read("eta.gpi"); \\ for Etax() and Etax_plus() \\ Definition according to Gasper, Rahman, "Basic hypergeometric series". \\ ++++++++++++++++++++++++++++++++++++++++ \\ q-exponentials, first kind: Exp_q(q, x, S=-1)={ Etax_plus(q, x, S); } exp_q(q, x, S=-1)={ 1 / Etax(q, x, S); } \\ \\ lim(q-->1, exp_q(q, (1-q)x)) = lim(q-->1, Exp_q(q, (1-q)*x) ) = exp(x) \\ \\ Exp_q(q, +x) * exp_q(q, -x) = 1 \\ Exp_q(q, -x) * exp_q(q, +x) = 1 \\ ( cf. exp(+x) * exp(-x) = 1 ) \\ Qderiv( Exp_q, q, x ) = Exp_q(q, x) / ((1-q)*(1+x)) \\ Qderiv( exp_q, q, x ) = exp_q(q, x) / (1-q) \\ Multiplicative splitting: \\ \\ exp_q(q, x) = prod( s=0, M-1, exp_q(q^M, q^s * x) ) for M>=1 \\ for example, \\ exp_q(q, x) = exp_q(q^2, x) * exp_q(q^2, q*x) \\ exp_q(q, x) = exp_q(q^3, x) * exp_q(q^3, q*x) * exp_q(q^3, q^2*x) \\ (And the same for Exp_q() ) \\ \\ Let W = exp(2*I*Pi/M), then \\ exp_q(q^M, x^M) = prod( s=0, M-1, exp_q(q, W^s * x) ) for M>=1 \\ (And the same for Exp_q() ) \\ Inversion of parameters: \\ exp_q(1/q, x) = Exp_q(q, -q * x) \\ Exp_q(1/q, x) = exp_q(q, -q * x) \\ exp_q(1/q, x/q) = Exp_q(q, -x) \\ Exp_q(1/q, x/q) = exp_q(q, -x) \\ exp_q(q, x/q) = q/(q - x) * exp_q(q, x) \\ Exp_q(q, x/q) = (q + x)/q * Exp_q(q, x) \\ exp_q(1/q, x*q) = 1 / ( (1 - q * x) * exp_q(q, +q*x) ) \\ Exp_q(1/q, x*q) = (1 + q * x) / Exp_q(q, +q*x) \\ \\ exp_q(q, 1/x) = x / (x - 1) * exp_q(q, q/x) \\ Exp_q(q, 1/x) = (x + 1) / x * Exp_q(q, q/x) \\ exp_q(q, 1/x) = (q*x - 1) / (q*x) * exp_q(q, 1/(q*x)) \\ Exp_q(q, 1/x) = q*x / (q*x + 1) * Exp_q(q, 1/(q*x)) \\ \\ exp_q(1/q, 1/x) = x / (x - 1) * exp_q(1/q, 1/(q*x)) \\ Exp_q(1/q, 1/x) = (x + 1) / x * Exp_q(1/q, 1/(q*x)) \\ \\ exp_q(q, q/x) = 1 / (1 - q/x) * exp_q(q, q^2/x) \\ exp_q(q, q/x) = 1 / ((1 - q/x) * (1 - q^2/x) ) * exp_q(q, q^3/x) \\ exp_q(q, q/x) = 1 / ((1 - q/x) * (1 - q^2/x) * (1 - q^3/x) ) * exp_q(q, q^4/x) \\ etc., similarly for Exp_Q(): \\ Exp_q(q, q/x) = (1 + q/x) * Exp_q(q, q^2/x) \\ Exp_q(q, q/x) = (1 + q/x) * (1 + q^2/x) * Exp_q(q, q^3/x) \\ Exp_q(q, q/x) = (1 + q/x) * (1 + q^2/x) * (1 + q^3/x) * Exp_q(q, q^4/x) \\ The following identities are essentially the same as \\ for Etax() and Etax_plus(): \\ \\ exp_q(+q, +x^2) = exp_q(+q, +x) * exp_q(+q, -x) * exp_q(+q^2, +q*x^2) \\ \\ exp_q(+q, -x^2) = 1/Exp_q(+q, +x^2) = \\ = exp_q(+q, +I*x) * exp_q(+q, -I*x) * exp_q(+q^2, -q*x^2) = \\ = 1/Exp_q(+q^2, +x^2) * exp_q(+q^2, -q*x^2) = \\ = 1/Exp_q(+q^2, +x^2) * 1/Exp_q(+q^2, +q*x^2) = \\ = exp_q(+q^2, -x^2) * exp_q(+q^2, -q*x^2) \\ \\ exp_q(+q, +I*x) * exp_q(+q, -I*x) = exp_q(+q^2, -x^2) \\ Exp_q(+q, +I*x) * Exp_q(+q, -I*x) = Exp_q(+q^2, +x^2) \\ \\ exp_q(+q, +x) * exp_q(+q, -x) = exp_q(+q^2, +x^2) \\ Exp_q(+q, +x) * Exp_q(+q, -x) = Exp_q(+q^2, -x^2) \\ \\ Jacobi's triple product identity: \\ Exp_q(q, +x) * Exp_q(q, +q/x) * Exp_q(q, -q) = \\ = sum(n=-inf..+inf, x^n * q^((n^2-n)/2) ) = \\ = 1 / ( exp_q(q, -x) * exp_q(q, -q/x) * exp_q(q, +q) ) \\ \\ Replace q by q^2 and x by q*x to obtain \\ Exp_q(q^2, +x*q) * Exp_q(q^2, +q/x) * Exp_q(q^2, -q^2) = \\ = sum(n=-inf..+inf, x^n * q^(n^2) ) = \\ = 1 / ( exp_q(q^2, -x*q) * exp_q(q^2, -q/x) * exp_q(q^2, +q^2) ) \\ \\ exp_q(+q, +q) = 1 / E1 \\ exp_q(+q, -q) = E1 / E2 \\ exp_q(-q, -q) = E1*E4 / E2^3 \\ exp_q(-q, +q) = E2^2 / E1*E4 \\ exp_q(+q^2, +q) = E2 / E1 \\ exp_q(+q^2, +q^2) = 1 / E2 \\ exp_q(+q^2, -q) = E1*E4 / E2^2 \\ exp_q(q^2, -q) * exp_q(q^4, +q^4) = E1 / E2^2 \\ exp_q(+q^3, +q) * exp_q(+q^3, +q^2) = E3 / E1 \\ exp_q(+q^3, -q) * exp_q(+q^3, -q^2) = E1*E6 / E2*E3 \\ exp_q(-q^3, +q) * exp_q(-q^3, -q^2) = E2^2*E3*E12 / E1*E4*E6^2 \\ exp_q(+q^4, +q) * exp_q(+q^4, +q^3) = E2 / E1 = exp_q(+q^2, +q) = \\ = exp_q(+q^8, +q) * exp_q(+q^8, +q^3) * exp_q(+q^8, +q^5) * exp_q(+q^8, +q^7) \\ \\ Exp_q(+q, +q) = E2 / E1 = 1 / exp_q(+q, -q) \\ Exp_q(+q, -q) = E1 \\ Exp_q(-q, +q) = E2^3 / E1*E4 = 1 / exp_q(-q, -q) \\ Exp_q(-q, -q) = E1*E4 / E2^2 \\ Exp_q(+q^2, +q) = E2^2 / E1*E4 \\ Exp_q(+q^2, +q) = E4 / E2 \\ Exp_q(+q^2, -q) = E1 / E2 \\ Exp_q(+q^2, +q) * Exp_q(+q^4, -q^4) = E2^2 / ( E1 ) \\ \\ Some expansions: \\ exp_q(q/(1-q), q/(1-q)) [cf. A103446] \\ Exp_q(q/(1-q), q/(1-q)) [cf. A129519] \\ \\ Let E = exp_q(q, q), then \\ q * deriv(E, 'q) / ( E ) = sum(n>=1, n * x^n/(1-x^n) ) [cf. A000203] \\ \\ Let E = Exp_q(q, q), then \\ q * deriv(E, 'q) / ( E ) = sum(n>=1, n * x^n/(1+x^n) ) = \\ = (Theta3(q)^4 + Theta2(q)^4 - 1)/24 [cf. A000593] \\ \\ ++++++++++++++++++++++++++++++++++++++++ \\ q-cosine, q-sine, q-tangent, first kind: cos_q(q, x, S=-1)={ ( exp_q(q, +I*x, S) + exp_q(q, -I*x, S) ) / 2; } sin_q(q, x, S=-1)={ ( exp_q(q, +I*x, S) - exp_q(q, -I*x, S) ) / (2*I); } Cos_q(q, x, S=-1)={ ( Exp_q(q, +I*x, S) + Exp_q(q, -I*x, S) ) / 2; } Sin_q(q, x, S=-1)={ ( Exp_q(q, +I*x, S) - Exp_q(q, -I*x, S) ) / (2*I); } \\ cos_q(q, x)*Cos_q(q, x) + sin_q(q, x)*Sin_q(q, x) = 1 \\ sin_q(q, x)*Cos_q(q, x) - cos_q(q, x)*Sin_q(q, x) = 0 \\ \\ Let C = cos_q(a, b) * Cos_q(a, b) and S = sin_q(a, b) * Sin_q(a, b). \\ For a=-q^2, b=+q we have C - S = E2^2*E8^7 / ( E4^7*E16^2 ) \\ = C^2 - S^2 (because C + S = 1) \\ C * S = q^2 * E2^4*E8^2*E16^4 / ( E4^10 ) \\ (C + I*S)^2 = E2^2*E8^7 / ( E4^7*E16^2 ) + 2*I*q^2* E2^4*E8^2*E16^4 / ( E4^10 ) \\ \\ For a=+q^2, b=+I*q we have C - S = E8^5 / ( E2^2*E4*E16^2 ) \\ C * S = -q^2 * E4^2*E16^4 / ( E2^4*E8^2 ) \\ (C + I*S)^2 = E8^5 / ( E2^2*E4*E16^2 ) - 2*I*q^2 * E4^2*E16^4 / ( E2^4*E8^2 ) tan_q(q, x, S=-1)={ sin_q(q, x, S) / cos_q(q, x, S); } cot_q(q, x, S=-1)={ cos_q(q, x, S) / sin_q(q, x, S); } \\ \\ tan_q(q, x) = Sin_q(q, x, S) / Cos_q(q, x, s); \\ cot_q(q, x) = Cos_q(q, x, S) / Sin_q(q, x, s); \\ \\ Let T = tan_q(a, b) and C = cos_q(a, b), then \\ for a=-q^2 and b=+q we have C - T = +1/q* E8^6 / ( E4^2*E16^4 ) \\ and C + T = +1/q* E4^5 / ( E2^2*E8*E16^2 ), \\ so C^2 - T^2 = +1/q^2* E4^3*E8^5 / ( E2^2*E16^6 ) \\ for a=+q^2 and b=-I*q we have C - T = +I * 1/q* E8^6 / ( E4^2*E16^4 ) \\ and C + T = +I * 1/q* E2^2*E8 / ( E4*E16^2 ), \\ so C^2 - T^2 = -1/q^2* E2^2*E8^7 / ( E4^3*E16^6 ) \\ ++++++++++++++++++++++++++++++++++++++++ \\ hyperbolic q-cosine, q-sine, q-tangent, first kind: cosh_q(q, x, S=-1)={ ( exp_q(q, +x, S) + exp_q(q, -x, S) ) / 2; } sinh_q(q, x, S=-1)={ ( exp_q(q, +x, S) - exp_q(q, -x, S) ) / 2; } Cosh_q(q, x, S=-1)={ ( Exp_q(q, +x, S) + Exp_q(q, -x, S) ) / 2; } Sinh_q(q, x, S=-1)={ ( Exp_q(q, +x, S) - Exp_q(q, -x, S) ) / 2; } tanh_q(q, x, S=-1)={ sinh_q(q, x, S) / cosh_q(q, x, S); } coth_q(q, x, S=-1)={ cosh_q(q, x, S) / sinh_q(q, x, S); } \\ \\ Let T = tanh_q(a, b) and C = coth_q(a, b), then \\ for a=+q^2 and b=+q we have C + T = +1/q* E8^6 / ( E4^2*E16^4 ) \\ and C - T = +1/q* E2^2*E8 / ( E4*E16^2 ) \\ so C^2 - T^2 = +1/q^2* E2^2*E8^7 / ( E4^3*E16^6 ) \\ and ( C + T ) / ( C - T ) = E8^5 / ( E2^2*E4*E16^2 ) \\ and (C^2 - T^2)*(C - T) = C^3 - T*C^2 - T^2*C + T^3 = \\ (using C*T=1) = C^3 - C - T + T^3 = 1/q^3* E2^4*E8^8 / ( E4^4*E16^8 ) \\ = C^3 + T^3 - (C + T) = C^3 + T^3 - (+1/q* E8^6 / ( E4^2*E16^4 )) \\ so C^3 + T^3 = +1/q* E8^6 / ( E4^2*E16^4 ) + 1/q^3* E2^4*E8^8 / ( E4^4*E16^8 ) \\ Now (C + T)^2 = 1/q^2*E8^12/(E4^4*E16^8) \\ = C^2 + 2*T*C + T^2 = C^2 + 2 + T^2, so \\ C^2 + T^2 = 1/q^2*E8^12/(E4^4*E16^8) - 2 \\ and (C - T)^2 = 1/q^2* E2^4*E8^2 / ( E4^2*E16^4 ) \\ = C^2 - 2*T*C + T^2 = C^2 - 2 + T^2, so \\ C^2 + T^2 = 1/q^2* E2^4*E8^2 / ( E4^2*E16^4 ) + 2 \\ so 1/q^2* E2^4*E8^2 / ( E4^2*E16^4 ) + 2 = 1/q^2*E8^12/(E4^4*E16^8) - 2 \\ this gives E2^4*E4^2*E8^2*E16^4 + 4*q^2*E4^4*E16^8 - E8^12, and finally \\ 0 = E1^4*E2^2*E4^2*E8^4 + 4*q*E2^4*E8^8 - E4^12 [Somos' t8_12_48] \\ ==== end of file ====