The Dissections Of Ramanujan Theta Function And Ramanujan Cubic Continued Fraction

Ramanujan theta functions and continued fractions are one of the improtant contents in the Ramanujan's lost notebook,It has been deeply studied by many scholars.In recent years,the study of their dissections is a hot topic.With the continuous development of various disciplines,the research methods of dissection are mainly Jacobi triple product identity,quintuple product identity,the linear combination of k theta functions and related theorem in[30]with Lewis.The paper is organized as following three parts.First,the 3-dissection of ?(q)are given by the linear combination of k theta functions.Then,the 2-dissection and 4-dissection of ?(q)are obtained by using the Jacobi triple product identity,and the 3-dissection of ?(-q)are obtained according to the equation in article[10,p.345,Entry 1],the 3-dissection of ?(-q)is simply generalized.Finally,we study Ramanujan-Selberg continued fraction and Ramanujan cubic continued fraction,and gain the 2-dissection and 4-dissection of the reciprocal of Ramanujan-Selberg continued fraction by using the Jacobi triple product identity.In 2010,M.D.Hirschhorn and Roselin obtained the 2-dissection and 4-dissection of Ramanujan cubic continued fraction's reciprocal by partial substitution method.In this paper,different forms of 2-dissection and 4-dissection of the reciprocal of Ramanujan cubic continued fraction are obtained by using the related theorem of Lewis in[30].