On existence of meromorphic solutions for certain q-difference equation

ScienceAsia(2023)

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Abstract
We consider the existence of transcendental meromorphic solutions of q-difference equation Sigma(n)(j =1) c(j)(z)f(q(j)z) = P (z, f (z))/Q(z, f (z)), where P(z, f (z)) and Q(z, f (z)) are polynomials in f having rational coefficients and no common roots, c(j)(z) are rational functions, q is an element of C and 0 < vertical bar q vertical bar <= 1. We obtain that such equation has no transcendental meromorphic solutions for the case m = deg(f) P - deg(f) Q >= 2.
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Key words
q-difference equation,difference equation,existence of transcendental meromorphic solution
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