A PROOF OF SELLERS’ CONJECTURE PETER PAULE AND SILVIU RADU Abstract. In 1994 James Sellers conjectured an infinite family of Ramanujan type congruences for 2-colored Frobenius partitions introduced by George E. Andrews. These congruences arise modulo powers of 5. In 2002 Dennis Eich- horn and Sellers were able to settle the conjecture for powers up to 4. In this article we prove Sellers’ conjecture for all powers of 5. 1. Introduction In his 1984 Memoir [1], George E. Andrews introduced two families of partition functions, φ k (m) and cφ k (m), which he called generalized Frobenius partition func- tions. In this paper we restrict our attention to 2-colored Frobenius partitions. Their generating function reads as follows [1, (5.17)]: (1) ∞  m=0 cφ 2 (m)q m = ∞  n=1 1 − q 4n−2 (1 − q 2n−1 ) 4 (1 − q 4n ) . In 1994 James Sellers [15] conjectured that for all integers n ≥ 0 and α ≥ 1 one has cφ 2 (5 α n + λ α ) ≡ 0 (mod 5 α ), where λ α is defined to be the smallest positive integer such that (2) 12λ α ≡ 1 (mod 5 α ). In his joint paper with Dennis Eichhorn [4] this conjecture was proved for the cases α =1, 2, 3, 4. In this paper we settle Sellers’ conjecture for all α in the spirit of G. N. Watson [16]. Several authors (e.g. [9], [2]) have stated that the method of Watson works well when the modular functions involved live on a Riemann surface of genus 0. The reason for this is that every such modular function can be written as a rational function (in Watson’s case polynomial function) in some fixed modular function t. In contrast to this, the modular functions that appear in this paper belong to a Riemann surface of genus 1. Treatments of this type are very rare in the literature. To the best of our knowledge only the papers by B. Gordon and K. Hughes [6], [7] and [8] apply Watson’s method to genus 1 Riemann surfaces. In these papers the authors use a relatively simple trick on the modular equation to make Watson’s method work for larger genus then 0. We are applying essentially the same idea in this paper; see Lemma 3.4 below. Our article is structured as follows. In Section 2 we state the Main Theorem (The- orem 2.7) of our paper. It describes the action of a class of Rademacher operators P. Paule was partially supported by grant P2016-N18 of the Austrian Science Funds FWF. S. Radu was supported by DK grant W1214-DK6 of the Austrian Science Funds FWF. 2010 Mathematics Subject Classification: primary 11P83; secondary 05A17. Keywords and phrases: generalized Frobenius partitions, Sellers’ conjecture, partition congru- ences of Ramanujan type. 1