The Generalization Of A Partition Function Equality And A New Proof Of Hypergeometric Functions Identity

Partition Function is one important part of q-series. With the development of q-series, people have gone deep to the study of partition function (If n is a positive integer, let P(n) denote the number of unrestricted representations of n as a sum of positive integers, where representations with different orders of the same summands are not regarded as distinct. We call P(n) the partition function). When referring to partition function, people will associated generalized function∑_n=0^∞P(n)qⁿ=∏_n=0^∞(1-qⁿ)^-1 of partition function P(n) which was firstly put forward by Euler.For example :3=3=2+1=1+1+1, P(3)=3; 5=5=1+4=1+1+3= 1+1+1+2=1+1+1+1+1=2+2+1=2+3, P(5)=7.This article contains four chief sections :In the first section, we define four partition functions : P_â–³m+c(n),_dP_â–³m+c(n),P_â–½m(n)å’Œ_dP(â–½m)(n).In the second section, we prove four partition functions identities:In the third section, we give some applications and qualities of these identities above.Finally, we give a new proof to the Hypergeometric functions identity Fⁿ(1-((?)⁴(-q))/((?)⁴(q))=F(1-((?)⁴(-qⁿ))/((?)⁴(qⁿ))...