Product of polynomial values at integral points and some of its applications

We are having an explicit asymptotic formula for the product P(1)P(2)P(3)…P(N), (*) where P(x) is a polynomial with complex coefficients. We use this to define a Gamma function Γ P for a polynomial P. Another application of the asymptotic formula for the product in (*) is the closed form evaluation of some infinite products. In fact, we evaluate ∏n≥1P(n)Q(n) where P(x)and Q(x) are polynomials in x, in terms of Gamma functions whenever the product is convergent. We apply this to evaluate some infinite products dealt by Ramanujan [7], Prudnikov [6], Borwein etc. [2], [3] and P. Abbott (entry (19) in [9]).We are having an explicit asymptotic formula for the product P(1)P(2)P(3)…P(N), (*) where P(x) is a polynomial with complex coefficients. We use this to define a Gamma function Γ P for a polynomial P. Another application of the asymptotic formula for the product in (*) is the closed form evaluation of some infinite products. In fact, we evaluate ∏n≥1P(n)Q(n) where P(x)and Q(x) are polynomials in x, in terms of Gamma functions whenever the product is convergent. We apply this to evaluate some infinite products dealt by Ramanujan [7], Prudnikov [6], Borwein etc. [2], [3] and P. Abbott (entry (19) in [9]).