On the Expansions of the Elliptic and Zeta Functions of ⅔K in Powers of q

J. W. L. Glaisher
1905 Proceedings of the London Mathematical Society  
I have given the expansions of the twelve elliptic and four zeta functions of \K in powers of q, the coefficients in the expansions being expressed by means of certain arithmetical functions. Since the publication of that paper I have reduced the number of these arithmetical functions, which are required for the expansions, to two. The new forms may be deduced from those contained in the previous paper, but it seems preferable to give an independent investigation, deriving them from the general
more » ... em from the general expansions of the elliptic and zeta functions. In the previous paper the results were stated without proof, the methods by which they were obtained being merely indicated. 2. The expansions in powers of q of the general elliptic and zeta functions may be written t kp sn px = 42^ A (sin mx) q im , kk'p sd px = 42r ( -D^-^A t e i n mx)q im , kp cd px = 42? E (cos mx) q*"\ kp en px = 4 i r (-l)^m~^E(cos mx) q im ; pzn px = 42r A' (sin 2nx)q n , pzdpx= 42r(-l)*A'(sin2wz)g w , p dn px = 1 + 4 2 ? E' (cos toix) q 11 , k'p nd px = l-|-42r (-l) n #'(cos 2nx)q n ; * " On the y-Series derived from the
doi:10.1112/plms/s2-2.1.340 fatcat:vrws34ivqfd2vgcl4nnfpmimuy