COMBINATORIAL AND MODULAR SOLUTIONS OF SOME SEQUENCES WITH LINKS TO A CERTAIN CONFORMAL MAP

PABLO ANDRES PANZONE

REVISTA DE LA UNIóN MATEMáTICA ARGENTINA

UNION MATEMATICA ARGENTINA

Lugar: Bahia Blanca; Año: 2018

0041-6932

Abstract. If fn is a free parameter, we give a combinatorial closed formsolution of the recursion(n + 1)2un+1 − fnun − n2un−1 = 0, n 1,and a related generating function. This is used to give a solution to the Ap´erytype sequencernn3 +rn−1nn3 −32 n2 +n2+ 2on − o+rn−2(n−1)3 = 0, n 2,for certain parameters , .We show from another viewpoint two independent solutions of the lastrecursion related to certain modular forms associated with a problem of conformalmapping: Let f() be a conformal map of a zero-angle hyperbolicquadrangle to an open half plane with values 0, , 1, 1 (0 < < 1) at thecusps and define t = t() := 1 f() f()−f()−1 . Then the functionE() = 12if0()f()11 − f()is a solution, as a generating function in the variable t, of the above recurrence.In other words, E() = r0+r1t+r2t2+. . . , where r0 = 1, r1 = −, = 2− 4 .