One answer to an open problem on the monotonicity of Gaussian hypergeometric functions with respect to parameters

作者:Bao, Qi*; Wang, Miao Kun; Zhang, Yu Ao
来源:Revista de la Real Academia de Ciencias Exactas Fisicas y Naturales Serie A-Matematicas, 2022, 116(3): 115.
DOI:10.1007/s13398-022-01258-w

摘要

The authors prove that for arbitrarily given b is an element of (0, infinity) r is an element of (0, 1) and n is an element of N-0, and for lambda is an element of R, the functions @@@ a bar right arrow F(a,b; a + b; r/a(lambda) and a bar right arrow F(a, b; a + b; r) - Sigma(n)(k=0) (a,k)(b,k)/(a+b,k)k! r(k)/a(lambda) @@@ are both strictly increasing (decreasing) on (0, infinity) if and only lambda <= 0 (lambda >= max{1, b}, respectively), where F (a, b; a + b; r) denotes the Gaussian hypergeometric function. This result gives one answer to an open problem raised by Qiu et al.