we consider gabor systems generated by a gaussian function and prove certain classical results of paley and wiener on nonharmonic fourier series of complex exponentials for the gabor expansion‎. ‎in particular, we prove a version of plancherel-po ́lya theorem for entire functions with finite order of growth and use the hadamard factorization theorem to study regularity‎, ‎exactness and deficienc...

Congruences and related identities are derived for a set of colored weighted partition functions whose generating generate the graded algebra integer weight modular forms level seven. The work determines general strategy identifying proving associated congruences on principal congruence subgroup [Formula: see text]. Ramanujan’s modulo text] serves as prototype process used to prove new