Generalized sine equations, I
Copyright policy )Generalized sine equations, I
[For part II see ibid. 49, No. 1-2, 122-152 (1995; Zbl 0839.39008).] The following is offered as main result. Functional equations of the form \(f(x + y) f(x - y) = P[f(x), f(y)]\), where \(P\) is a polynomial of degree less than 6, have solutions \(f\) over the reals which assume at least 86 distinct complex values and which are nonzero at 0 if, and only if, \(P\) is of one of three explicitly given forms, two quadratic, one biquadratic. The proof is lengthy and technical. However, a MATHEMATICA program is offered for verification, which may work also for polynomials of higher (but not too high) degrees.
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quadratically closed field, solution by computer program, polynomial, polynomials, functional equations, MATHEMATICA, algebraically closed field, Functional equations for complex functions, abelian group, Symbolic computation and algebraic computation, Article, exponential, trigonometric and hyperbolic functions, 510.mathematics, Functional equations for real functions, general solution, generalized sine equations, Functional equations for functions with more general domains and/or ranges, Wilson functional equations, functional equation
quadratically closed field, solution by computer program, polynomial, polynomials, functional equations, MATHEMATICA, algebraically closed field, Functional equations for complex functions, abelian group, Symbolic computation and algebraic computation, Article, exponential, trigonometric and hyperbolic functions, 510.mathematics, Functional equations for real functions, general solution, generalized sine equations, Functional equations for functions with more general domains and/or ranges, Wilson functional equations, functional equation
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