Bernoulli Numbers and a New Binomial Transform Identity

Transcription

1 Joural of Iteger Sequece, Vol , Article Beroulli Nuber ad a New Bioial Trafor Idetity H. W. Gould Departet of Matheatic Wet Virgiia Uiverity Morgatow, WV USA gould@ath.wvu.edu Jocely Quaitace Departet of Matheatic Rutger Uiverity Picataway, NJ USA quaita@ath.rutger.edu Abtract Let b 0 be the bioial trafor of a 0. We how how a bioial traforatio idetity of Che prove a yetrical Beroulli uber idetity attributed to Carlitz. We the odify Che idetity to prove a ew bioial traforatio idetity. Carlitz [1] poed a a proble the rearable yetric Beroulli uber idetity 1 B + 1 B +, 1 valid for arbitrary, 0. The publihed olutio by Shao [2] ued atheatical iductio o ad. The idetity wa redicovered recetly by Vailev ad Vailev- Miaa [10], but tated i the for B B +, 2

2 valid for arbitrary poitive iteger ad. Idetity 2 i equivalet to Idetity 1 ice [ 1 1 ]B + 0. Their proof ued the yetry of a fuctio f x,y ivolvig Beroulli uber itroduced i a eparate paper [9]. They give o referece to Carlitz or to Shao proof. A alterative proof of Equatio 1 i derived through a applicatio of a bioial traforatio idetity dicovered by Che [3]. Let a be ay equece of uber, ad defie the bioial trafor of a to be the equece b, where b a. A corollary of [3, Th. 2.1] i a + 1 b +. 3 The Beroulli uber atify the recurrece B 1 B for 0. Settig a B, we the have b 1 B, o that Equatio 3 becoe B B +, which i preciely Idetity 1 of Carlitz. Che proof of Equatio 3 relie o certai propertie of Seidel atrice. We preet a direct proof which relie o the hypergeoetric idetity x+ 1 r x 1 ; 4 r ee [6, Idetity 3.47, p. 27]. I Equatio 4 we require that ad r be oegative iteger ad x be a coplex uber. Sice the bioial trafor ivert to give a 1 b we fid that a b + 1 b b b 1 b +. A careful aalyi of thi precedig proof yield a hort proof of [3, Th. 3.2], where Che relie o legthy iductio arguet. We will itead ue Equatio 4. 2

3 Theore 1. [3, Th. 3.2] Let b be the bioial trafor of a. The + + a + 1 b +, 5 for arbitrary oegative,, ad. Proof. By defiitio b a. Thi iplie that a 1 b. Hece a b b b b b +, where the fourth equality follow by Equatio 4. Equatio 5 allow u to etablih a geeralizatio of the curiou forula + + x 1 1+x, 6 dicovered by Sio [8]. A quic proof of thi wa give by Gould [7] uig eleetary propertieoflegedrepolyoial. Itead,chooea x forall 0. Theb 1+x ad Idetity 5 tell u that + x Lettig recover Idetity 6. Through a iductio arguet Che prove i0 1+x +. Theore 2. [3, Th. 3.1] Let b be the bioial trafor of a. The ++ a b i a + i ++1+i ++i, 7 where,, ad are oegative iteger. 3

4 If we ue Equatio 4 ad the followig hypergeoetric idetity attributed to Frich [4], [5, p. 337], 1 1 c 1, b c > 0, 8 +c b+ c [6, Idetity 4.2, p. 46], we are able to prove the followig ew bioial traforatio idetity. Theore 3. Let b be the bioial trafor of a. Let,, ad be oegative iteger. The a b Proof. By defiitio b a. Hece a 1 b ad a b b b b b b b b b b b b b. The fourth lie follow fro Equatio 4 while the eveth follow fro Equatio 8. I uary, we have how that a b b, b b c b

5 If we copare Idetity 7 to Idetity 10, we coclude that b i a i ++1+i ++i. 11 i0 Equatio 11 ca be furthered iplified by applyig Equatio 8. I particular, 1 i0 1 1 i i a ++1+i ++i a i i i0 +1 a Thee calculatio how that Equatio 11 i equivalet to a 1 Set +1 to obtai a a i b b Sice , we ee that Equatio 13 i equivalet to Equatio 9. Referece [1] L. Carlitz, Proble 795, Math. Mag , 107. [2] A. G. Shao, Solutio of Proble 795, Math. Mag , [3] K. W. Che, Idetitie fro the bioial trafor, J. Nuber Theory ,

6 [4] R. Frich, Sur le ei-ivariat et oet eployé da l étude de ditributio tatitique, Srifter utgitt av Det Nore Videap-Aadei i Olo, II. Hitori- Filoofi Klae, 1926, No. 3, 87 pp. [5] Euge Netto, Lehrbuch der Cobiatori, 2d editio, Reprited by Chelea, [6] H. W. Gould, Cobiatorial Idetitie, A Stadardized Set of Table Litig 500 Bioial Coefficiet Suatio, revied editio. Publihed by the author, Morgatow, WV, [7] H. W. Gould, A curiou idetity which i ot o curiou. Math. Gaz , 87. [8] S. Sio, A curiou idetity, Math. Gaz , [9] P. Vailev ad M. Vailev-Miaa, O the u of equal power of the firt ter of a arbitrary arithetic progreio, Note o Nuber Theory ad Dicrete Matheatic , [10] P. Vailev ad M. Vailev-Miaa, O oe rearable idetity ivolvig Beroulli uber, Note o Nuber Theory ad Dicrete Matheatic , Matheatic Subect Claificatio: Priary 11B68; Secodary 05A10, 11B65. Keyword: Beroulli uber, bioial trafor. Cocered with equece A ad A Received October ; revied verio received Jauary Publihed i Joural of Iteger Sequece, Jauary Retur to Joural of Iteger Sequece hoe page. 6