2023/10/16 by Anne-Maria Ernvall-Hytönen, Ernvall-Hytönen, Anne-Maria, Tapani Matala-aho +1
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2310.11468
openalex publication_date 2023/10/16 · openalex created_date 2023/10/21 · openalex updated_date 2026/07/28
We are interested in finding an explicit estimate to the binomial sum Qn(x)=∑k=0n k! n\choose k2 (-x)k at x=1 for n=0,1,2,…. Despite of its own interest the polynomial Qn(x) is important as the denominator in the Padé identity of the Euler's factorial series E(x) = ∑k=0∞ k! xk as well as its close connection to a classical Laguerre polynomial Ln(x) = (1)/(n!) ex ((d)/(dx))n (e-xxn). Our main result is the explicit bound |Ln(1)-√\fraceπ⋅ \fraccos (2√(n)-\fracπ4)n1/4 +(17)/(48)√\fraceπ\fracsin(2√(n)-\fracπ4)n3/4|lt;(0.51)/(n) for all n=0,1,2,…, which replaces the Fejér's asymptotic formula from 1909. As a corollary of this, one also gets a new proof for the bound |Qn(1)| ≤ n!, and even more.