Further Equivalent Binomial Sums

Abstract

Five binomial sums are extended by a free parameter m, that are shown, through the generating function method, to have the same value. These sums generalize the ones by Ruehr (1980), who discovered them in the study of two unexpected equalities between definite integrals. 2020 Mathematics Subject Classification. 11B65, 05A10. Manuscript received 22nd December 2020, revised and accepted 2nd February 2021. In 1980, Kimura [14] proposed a monthly problem about two curious identities of definite integrals. If f is continuous on [− 1 2 , 3 2 ], then for δ= 0, 1, prove that ∫ 3 2 − 2 x f ( 3x −2x3)dx = 2∫ 1 0 x f ( 3x −2x3)dx. In his (trigonometric) proof, Ruehr [14] observed by linearity that to prove these identities, it is enough to verify them for monomials f (x) = xn . This led him to discover the following interesting identities An =Cn and Bn = Dn , where for a natural number n, the four binomial sums are defined by An = n ∑ j=0 3 j ( 3n − j 2n ) , Bn = n ∑ j=0 2 j ( 3n +1 2n + j +1 ) , Cn = 2n ∑ j=0 (−3) j ( 3n − j n ) , Dn = 2n ∑ j=0 (−4) j ( 3n +1 n + j +1 ) . ∗Corresponding author. ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/ 422 Mei Bai and Wenchang Chu Allouche [1, 2] examined the related integrals and reviewed these identities in a more elegant manner. These identities were also reconfirmed by Meehan et al [16] who found, through the WZ-algorithm, that these four sequences satisfy also the common recurrence relation: X0 = 1 and Xn+1 = 27 4 Xn − 3 (3n+1 n ) 4(n +1) . By introducing a variable, Alzer and Prodinger [3] recently considered the polynomial analogues, that were also examined by Kilic–Arikan [13] through bijections. By applying the generating function approach to the binomial convolutions Ωn = n ∑ k=0 ( 3k k )( 3n −3k n −k ) the authors [4] not only confirmed the identities Ωn = An = Bn =Cn = Dn ; (1) but also found the two additional ones Ωn = En = Fn , (2)