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Since the study by Jacobi and Hecke, Hecke-type series have received extensive attention. Especially, Hecke-type series involving infinite products have attracted broad interest among many mathematicians including Kac, Peterson, Andrews, Bressoud and Liu. Motivated by the works of these people, we study Hecke-type series involving infinite products. In particular, we establish some Hecke-type series involving infinite products and then obtain the truncat
Since the study by Jacobi and Hecke, Hecke-type series have received a lot of attention. Unlike such series associated with indefinite quadratic forms, identities on Hecke-type series associated with definite quadratic forms are quite rare in the lit
In an additive group (G,+), a three-dimensional corner is the four points g, g+d(1,0,0), g+d(0,1,0), g+d(0,0,1), where g is in G^3, and d is a non-zero element of G. The Ramsey number of interest is R_3(G) the maximal cardinality of a subset of G^3 t
Let $R$ be a finite ring and define the hyperbola $H={(x,y) in R times R: xy=1 }$. Suppose that for a sequence of finite odd order rings of size tending to infinity, the following square root law bound holds with a constant $C>0$ for all non-trivial
We give unified modular proofs to all of Gospers identities on the $q$-constant $Pi_q$. We also confirm Gospers observation that for any distinct positive integers $n_1,cdots,n_m$ with $mgeq 3$, $Pi_{q^{n_1}}$, $cdots$, $Pi_{q^{n_m}}$ satisfy a nonze
The Hecke orbit conjecture asserts that every prime-to-$p$ Hecke orbit in a Shimura variety is dense in the central leaf containing it. In this paper, we prove the conjecture for certain irreducible components of Newton strata in Shimura varieties of