2023/03/09 by Hailu Bikila Yadeta, Yadeta, Hailu Bikila
Mathematics · #05A10 #05A20. 26A18 #26A42 #26D07 #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Inequalities and Applications #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.2303.05426
openalex publication_date 2023/03/09 · openalex created_date 2023/03/12 · openalex updated_date 2026/07/28
In this paper, we derive some new combinatorial inequalities by applying well known real analytic results like Hölder's inequality, Young's inequality, and Minkowiski's inequality to the recursively defined sequence fn of functions f0(x) amp; = χ(-1/2, 1/2) (x), fn+1(x) amp; = fn(x+1/2)+ fn(x-1/2), n ∈ ℕ ∪ \0\. Towards this goal, we derive the closed form of the aforementioned sequence (fn)_n∈ ℕ ∪ \0\ of functions and show that it is a sequence of simple functions that are linear combinations of characteristic functions of some unit intervals In,i, i=0,1, ..., n , with values the binomial coefficients \binomni on each unit interval In,i. We show that fn ∈ Lp(ℝ)), 1≤ p ≤ ∞ . Besides applying real analytic methods to formulate some combinatorial inequalities, we also illustrate the application of some combinatorial identities. For example, we use the Vandermonde convolution (or Vandermonde identity), in the study of some properties of the sequence of functions (fn)_n∈ℕ∪ \0\. We show how the L2 norm of fn is related to the Catalan numbers.