2025/02/26 by S. Ananya, C Channabasavayya, Ananya, S. +5 · 1 citation
Decision Sciences · Engineering · #Combinatorics (math.CO) #FOS: Mathematics #Multi-Criteria Decision Making #Number Theory (math.NT) #Optimization and Mathematical Programming
paper · pdf · doi:10.48550/arxiv.2503.11670
openalex publication_date 2025/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the recent past, the work in the area of vanishing coefficients of infinite q-products has been taken to the forefront. Weaving the same thread as Ramanujan, Richmond, Szekeres, Andrews, Alladi, Gordon, Mc Laughlin, Baruah, Kaur, Tang, we further prove vanishing coefficients in arithmetic progressions moduli 5, 7, 11, 13, 19, 21, 23 and 29 of the following four families of infinite products, where \Xa,b,sm,km,u,v(n)\n≥ n0, \Ya,b,sm,km,u,v(n)\n≥ n0, \Za,b,sm,km,u,v(n)\n≥ n0 and \Wa,b,sm,km,u,v(n) \n≥ n0 are defined by ∑n≥ n0∞Xa,b,sm,km,u,v(n)qn:=amp;(qa,qsm-a;qsm)inftyu(qb,qkm-b;qkm)inftyv,
∑n≥ n0∞Ya,b,sm,km,u,v(n)qn:=amp;(qa,qsm-a;qsm)inftyu(-qb,-qkm-b;qkm)inftyv,
∑n≥ n0∞Za,b,sm,km,u,v(n)qn:=amp;(-qa,-qsm-a;qsm)inftyu(qb,qkm-b;qkm)inftyv,
∑n≥ n0∞Wa,b,sm,km,u,v(n)qn:=amp;(-qa,-qsm-a;qsm)inftyu(-qb,-qkm-b;qkm)inftyv, here a, b, s, k, u and v are chosen in such a way that the infinite products in the right-hand side of the above are convergent and n0 is an integer (possibly negative or zero) depending on a, b, s, k, u and v. The proof uses the Jacobi triple product identity and the properties of Ramanujan general theta function.