Analytic Aspects of the Generalized Second-Order Central Trinomial Coefficients

Yu-Yang Zhang (), Hao Pan and Lei Chen
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Yu-Yang Zhang: School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing 210023, China
Hao Pan: School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing 210023, China
Lei Chen: School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing 210023, China

Mathematics, 2025, vol. 13, issue 9, 1-13

Abstract: The generalized second-order central trinomial coefficients T n ( 2 ) ( b , c ) are a special case of the generalized r -order central trinomial coefficients, corresponding to r = 2 . We show that all zeros of the polynomial T ( 2 ) n ( x ) = ∑ k = 0 ⌊ n / 2 ⌋ T ( 2 ) ( n , k ) x k are real. And zeros of T ( 2 ) n − 1 ( x ) interlace those of T ( 2 ) n − 2 ( x ) , as well as those of T ( 2 ) n ( x ) . Using this result, we also discuss the asymptotic normality of T ( 2 ) ( n , k ) .

Keywords: generalized second-order central trinomial coefficients; real zeros; interlace; asymptotic normality (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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