Research Article Sums of Products of Cauchy Numbers, Including Poly-Cauchy Numbers

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1 Hiawi Publishig Corporatio Joural of Discrete Matheatics Volue 2013, Article ID , 10 pages Research Article Sus of Proucts of Cauchy Nubers, Icluig Poly-Cauchy Nubers Taao Koatsu Grauate School of Sciece a Techology, Hirosai Uiversity, Hirosai , Japa Correspoece shoul be aresse to Taao Koatsu; oatsu@cc.hirosai-u.ac.jp Receive 24 July 2012; Accepte 24 October 2012 Acaeic Eitor: Gi Sag Cheo Copyright 2013 Taao Koatsu. This is a ope access article istribute uer the Creative Coos Attributio Licese, which perits urestricte use, istributio, a reprouctio i ay eiu, provie the origial wor is properly cite. We ivestigate sus of proucts of Cauchy ubers icluig poly-cauchy ubers: T ) ) i 1 i, i 1,...,i 0 i 1,...,i ) c i1 c i 1 c ) i 1, 0).ArelatioaogthesesusT ) ) show i the paper a eplicit epressios of sus of two a three proucts the case of 2a that of 3escribeithepaper)aregive.Wealsostuytheotherthreetypesof sus of proucts relate to the Cauchy ubers of both is a the poly-cauchy ubers of both is. 1. Itrouctio The Cauchy ubers of the first i) c are efie by the itegral of the fallig factorial: 1 c 0 1 1) 1)! ) 1) 0 see [1, Chapter VII]). The ubers c /! are soeties calle the Beroulli ubers of the seco i see e.g., [2, 3]). Such ubers have bee stuie by several authors [4 8] because they are relate to various special cobiatorial ubers, icluig Stirlig ubers of both is, Beroulli ubers, a haroic ubers. It is iterestig to see that the Cauchy ubers of the first i c have the siilar properties a epressios to the Beroulli ubers B.For eaple, the geeratig fuctio of the Cauchy ubers of the first i c is epresse i ters of the logarithic fuctio: l 1) c!, 2) see [1, 6]), a the geeratig fuctio of Beroulli ubers B is epresse i ters of the epoetial fuctio: 0 e 1 B!, 3) 0 see [1]) or 1 e B 4) 0! see [9]). I aitio, Cauchy ubers of the first i c ca be writte eplicitly as c 1) 0 [ ] 1) 1 see [1, Chapter VII], [6, page 1908]), where [ ] are the usige) Stirlig ubers of the first i, arisig as coefficiets of the risig factorial 1) 1) [ ] 6) 0 see e.g., [10]). Beroulli ubers B i the latter efiitio) ca be also writte eplicitly as B 1) 0 5) { } 1)! 1, 7) where { } are the Stirlig ubers of the seco i, eterie by { } 1! j0 1) j j ) j) 8)

2 2 Joural of Discrete Matheatics see, e.g., [10]). Recetly, Liu et al. [5]establishesoerecurrecerelatiosaboutCauchyubersofthefirstias aalogous results about Beroulli ubers by Agoh a Dilcher [11]. I 1997 Kaeo [9] itrouce the poly-beroulli ubers B ) 0, 1)bythegeeratigfuctio where Li 1 e ) 1 e Li z) B ) 0 z 0!, 9) 10) is the th polylogarith fuctio. Whe 1, B 1) B is the classical Beroulli uber with B 1) 1 1/2. O the other ha, the author [12] itrouce the poly-cauchy ubers of the first i) c ) as a geeralizatio of the Cauchy ubers a a aalogue of the poly-beroulli ubers by the followig: c ) ) 1 2 1) 1 2 1) ) I aitio, the geeratig fuctio of poly-cauchy ubers is give by where Lif l 1)) 0 c ) 0 z!, 12) Lif z) :!1) 13) is the th polylogarith factorial fuctio, which is also itrouce by the author [12, 13]. If 1,thec 1) c is the classical Cauchy uber. The followig ietity o sus of two proucts of Beroulli ubers is ow as Euler s forula: i )B ib i B 1 1) B 0). 14) i0 The correspoig forula for Cauchy ubers was iscovere i [8]: i0 i )c ic i 1) c 2) c 1 0). 15) I this paper, we shall give ore aalogous results by ivestigatig a geeral type of sus of proucts of Cauchy ubers icluig poly-cauchy ubers: i 1 i i 1,...,i 0 )c i 1,...,i i1 c i 1 c ) i 1, 0), 1 16) whose Beroulli versio is iscusse i [14]. A relatio aog these sus a eplicit epressios of sus of two athreeprouctsarealsogive. 2. Mai Results We shall cosier the sus of proucts of Cauchy ubers icluig poly-cauchy ubers. Kaao [14] ivestigate thefollowigtypesofsusofproucts: S ) ) : i 1 i i 1,...,i 0 )B i 1,...,i i1 B i 1 1 B ) i 1, 0), 17) where Beroulli ubers B are efie by the geeratig fuctio 3) a poly-beroulli ubers B ) are efie by the geeratig fuctio 9) ali z) is the th polylogarithfuctioefiei10). It is show [14]that l0 1) l [ 1 l1 ]S l) 1 )! { [ 18) )! l ]B) l, ), { l0 { 0, 0 1). Cosier a aalogous type of sus of proucts of Cauchy ubers icluig poly-cauchy ubers: T ) ) : )c i i 1 i 1,...,i i1 c i 1 c ) i 1 i 1,...,i 0 19) Theweshowthefollowigresult. 1, 0). Theore 1. For a iteger a a oegative iteger,oe has l0 1) l [ 1 l1 ]T l) 1 )! { { l0 i0 i! l i ){ l }c) li ), { 0 0 1). 20) Note that the geeratig fuctio of T ) 1 l 1) ) Lif l 1)) Put Sice G ) : Lif l 1)) T ) 0 c ) 0 Lif 1 z) ez 1, Lif z 0 z) e z, Lif 1 z) z1) e z, is give by )!. 21)!. 22) 23)

3 Joural of Discrete Matheatics 3 we have Sice G 1 ) l 1), G 0 ) 1, G 1 ) 1)l 1) 1). l G l ) the coefficiet of i is equal to i0 c ) i li i! 24),l 0, 1), 25) l l G ) 26) c ) l ), )! 0 0 1). 27) We ee the followig lea i orer to prove Theore 1. Lea 2. For a iteger a a positive iteger,oehas i1 { i } 1 1) i i i )G ) 1 1) l 1)) 1) l [ 1 l1 ]G l ). Proof of Lea 2. Sice Lif ) l0!1)!1) 1!1) ) 28) 29) Lif 1 ) Lif ), we have G ) G 1 ) G ) 1) l 1). 30) By iuctio, we ca show that for 1 where G ) g ν1 ) : ν l0 1) ν 1) log 1)) ν [ ν ]g ν1 ), ν1 31) 1) ν l [ ν1 l1 ] G l ) ν1,2,...,). 32) Thus, by usig the iversio relatioship jν 1) j ν { j }[j ν ) ; ]{1 ν 0 ν1,2,..., 1) 33) see e.g., [10, Chapter 6]), the left-ha sie of the ietity i the previous lea is equal to j1 j { j } 1 1) j ν 1) j 1) j l 1)) ν [j ν ]g ν1 ) ν1 1 g 1) ν1 ) l 1)) ν 1) j ν { j }[j ν ] ν1 jν g 1 ) 1) l 1)), 34) which is the right-ha sie of the esire ietity. Now, by the geeratig fuctio 21), the ietity 25), a Lea 2, 0 l0 1) l [ 1 l1 ]T l) 1 ))! l 1) ) 1) l [ 1 l1 ]G l ) l0 i1 l0 1) i { i } i i )G ) 1) l { l } l l0 κ0 l0 0 c ) i li i0 i! l κ )κ { l } i0 l0 c ) i li i0 i! l i ){ l } c) li! i0 i! i! l i ){ l }c) li )!. Note that l ν ) 0l <ν)a{ 0 } 0 1). Therefore, l0 1) l [ 1 l1 ]T l) 1 ) 35)! { { l0 i0 i! l i ){ l }c) li ) ; { 0 0 1). 36) If we put 1i Theore 1, we get a aalogous forula to Euler s forula 14) for sus of proucts of Cauchy uber a a poly-cauchy uber.

4 4 Joural of Discrete Matheatics Corollary 3. Oe has i0 i )c i c 1) i c ) i ) 1) c) 1 c) 0). 37) 2.1. Eplicit Forula for T ) 2 ). Theore 1 gives oly relatios aog sus of proucts T ) ).For2, a eplicit forula for is give. T ) 2 ) i0 Theore 4. For 0a 1oe has T 0) 2 ) c 1) 1) 1) i0 T ) i 2 ) c 1) 1) T ) 2 ) c 1) where c 1) 1) c c 1. Proof. Cosier T 0) 2 ) 0! i )c ic ) i 38) 1) [ ] i ) 1 39) i1, j1 1 1) j0 c j) c j) l 1) Lif 0 l 1)) 1) l 1) c 1) 0 1) c j) 1 ), 40) 1) c j) 1 ), 41) 1)!, 42) where c ) z) are poly-cauchy polyoials of the first i, efie by the geeratig fuctio c ) Lif l 1)) 1) z c ) 0 z)!. 43) z) are epresse eplicitly i ters of the Stirlig ubers of the first i [13,Theore1]: c ) z) 0 [ ] 1) i ) z) i. 44) i0 i1) Hece, the ietity 39)holsbecause T 0) 2 ) c 1) 1) 1) i0 0 i [ ] 1) i ) 1 i0 i1 1) [ ] i ) 1 i1. 45) Net, by 30)aG 0 ) 1 we have G G j 1 ) G j ) j ) 1) l 1) j1 Hece, T ) 2 ) 0! j1 1 l 1) G ) 1) l 1). l 1) G ) 1) l 1) 1) G j ) c 1) 0 c 1) 0 0 j1 1)! 1) j1 1)! j1 c j) c j) )! 1) c j) 1 )!. 47) Therefore, we get the ietity 40). Fially, by Lif ) Lif 1 ) Lif ), 48) we have 1 1 j0 G G j 1 ) G j ) j ) j0 1) l 1) 49) G ) 1) l 1) 1 l 1). Hece, T ) 2 ) 0! l 1) G ) 1 1) l 1) 1) G j ) c 1) 0 c 1) 0 0 j0 1) 1! 1) j0 1)! 1 j0 c j) Therefore, we get the ietity 41). c j) 1 0! 1) c j) 1 )!. 50)

5 Joural of Discrete Matheatics 5 Table c 1 1/2 1/6 1/4 19/30 9/4 863/ / / /20 1) 1 3/2 5/6 1/4 11/30 11/12 271/84 117/8 7297/ /4 c 1) Puttig 1 i 40), we have the followig ietity, which is also fou i [8].Thisisalsoaaalogousforula to Euler s forula 14). Corollary 5. Oe has see also Table 1) i0 i )c ic i 1) c 2) c 1 0). 51) Proof. Sice T ) 3 ) T 0) 3 ) 2 1)c 1 1) c 2 ) Proof. Cosier 2 1) j0 2) 2 c j) 2 2 j 1 1) 2 3) c j) 1 c j) ). 57) T 0) 2 ) 0! 1) l 1) 1) c! 0 c c 1 )!, 0 52) T 0) 3 ) 0! 2 1) l 1)) 2 1) 0 T 1) 2 ) 0! T 1) 2 ) T 1) 2 1))!. 58) we have c 1) 1) c c 1.Hece, i0 i )c ic i T 1) 2 ) c 1) 1) c 1) c 1 ) 1) c 2) c 1. 53) 2.2. Eplicit Forulae for T ) 3 ). For 3, a eplicit forula for T ) 3 ) i0 i! i!j! i j)! c ic j c ) j0 i j 54) Hece, by Corollary 5 T 0) 3 ) T 1) 2 ) T 1) 2 1) 1) c 2) c 1 2) c 1 1) 3) c 2 ) 1) c 2 2) c 1 1) 3) c 2, 59) which is the ietity 55). By Lea 2 with 2 is also give. Theore 6. For 0a 1oe has T 0) 3 ) 1) c 2 2) c 1 1) 3) c 2, T ) 3 ) T 0) 3 ) 1 2 )c 1 1) c 2 ) 55) Thus, )G ) 2G ) G 1 )) 2G 1 ) G 2 )) 1) 2 l 1)) 2. 60) 1) j1 1 2 j 1 ) 2) 2 c j) 2 2 3) cj) 1 cj) ), 56) l j )G j ) 2G l ) G l 1 ) 1) 2 l 1)) 2 2G 0 ) G 1 ) 1) 2 l 1)) 2. 61)

6 6 Joural of Discrete Matheatics By ultiplyig both sies by 2 l 1 a suig over l fro 1 to,weobtai 2 G ) 1) 2 l 1)) 2 G 0 ) 1) 2 l 1)) 2 2 l 1 ) 2G 0 ) G 1 ) 1) 2 l 1)) 2 l1 2 l 1 l j l1 )G j ) 21 1)G 0 ) 2 1)G 1 ) 1) 2 l 1)) 2 l 2 j0 Fially, sice l 2 j0 l 2 j0 we obtai 1 )G j ) 2G j ) G j 1 ) 2G j 1 ) G j 2 ) 1) 2 l 1)) 2 2G 0 ) G 1 1) 2 l 1)) 2 2G l1 ) G l 1) 2 l 1)) 2 ), 65) j1 lj Hece, we have 2 2 l 1) ) G ) 2 l 1 ) )G j ) ) 2 1) l 1)) 2 62) 2 G ) 1) 2 l 1)) )G 1 ) )G 0 ) 1) 2 l 1)) 2 2 j0 Hece, we have lj2 2 l ) )G j ). 66) 2 1) 1) 2 l 1) 1) l 1)) 2 2 1) j 1 ) 2 j By coparig the coefficiets of /! i both sies, )G j ). 2 T ) 3 ) 2 1 1)T 0) 3 ) 2 1)T 0) 2 1) 2 1)T 0) 3 ) 2 2 j 1 ) j1 63) 2 2 l 1) ) G ) 1 2 ) 1) 2 1) 2 l 1) 2 l 1)) 2 2 1) j 1 2 ) j yielig the ietity 57). )G j ), 67) 1) c j) 2 1) 2) c j) 1 1) 2) 3) c j) 2 1) cj) 1 1) 2) c j) 2 ) 2 T 0) 3 ) 2 1)T 0) 2 1) j1 2 2 j 1 ) 1) 2) 2 c j) 2 2 3) cj) 1 cj) ). 64) Diviig both sies by 2, we have the ietity 56) Poly-Cauchy Nubers of the Seco Ki. Siilarly, efie T ) ) by T ) ) : i 1 i i 1,...,i 0 ) c i 1,...,i i1 c i 1 1 c ) i 1, 0), 68) where c ) is poly-cauchy uber of the seco i [12], whose geeratig fuctio is give by Lif l 1)) c ) 0!. 69)

7 Joural of Discrete Matheatics 7 Table c 1 1/2 5/6 9/4 251/30 475/ / /24 c 1) 1) 1 3/2 23/6 55/4 1901/ / / /8 c ) c ) cabealsoefieby ) 1 2 1) 1 2 1) ) see [12]). I this sese, c ) is calle poly-cauchy uber of the first i. Whe 1, c c 1) is the classical Cauchy uberofthesecoi,whosegeeratigfuctiois give by 1) l 1) c!. 71) By usig the correspoig lea to Lea 2, where G ) is replace by G ) Lif l1 )),wecaobtai the followig result. Theore 7. For a iteger a a oegative iteger,oe has l0 1) l [ 1 l1 ] T 1) 1 ) 1) 1 1) i! { l0 i0 i! l i 1 i ){ l } c) li! { )! c) ) ; { 0 0 1). 72) 0 Puttig 1i Theore 7,oehasthefollowig. Corollary 8. i0 i ) c i c 1) i c ) i ) c) 0). 73) Cosier the case 2.Notethatthegeeratig fuctio of poly-cauchy polyoial of the seco i z) [13]isgiveby c ) c ) 1) z Lif l 1)) c ) 0 z)!. 74) z) are epresse eplicitly i ters of the Stirlig ubers of the first i [13,Theore4] c ) z) [ ] 1) i ) z) i. 75) 0 i0 i1) Hece, c 1) 1) [ ] 1) i ) 1 0 i0 i1 1) [ ] i ) 1 i0 i i1. O the other ha, Thus, see Table 2). c 1) 0 1)! 1) 2 l 1) 1 c 1! μ0 0 ) μ ν ) c ν ν! ) c 1) 0 ν0 1)! i! 1)i c i!. i0 76) 77)! 1) 1) i! 1)i c i 78) i0 Theore 9. For 0a 1oe has T 0) 2 ) c 1) 1), 79) T ) 2 ) c 1) 1) c j), 80) T ) j1 2 ) c 1) 1 1) c j). 81) Puttig 1i 80), we have the followig ietity. This is also a aalogous forula to Euler s forula 14). Corollary 10. Oe has i0 j0 i ) c i c i c 1) 1) c 0). 82) If 3,the T ) 3 cabeepresseasfollows.

8 8 Joural of Discrete Matheatics Theore 11. For 0a 1oe has T 0) 3 ) 1) 1) l! l! l0 T 1) 2 l) 1) 1) l! l! c1) l 1) l c l ), l0 T ) 3 ) T 0) 3 ) 1 2 ) c 1) 1 2) 1) c j) j1 1 2 j 1 ) 2 1) 2)! l0 1) T ) 3 ) T 0) 3 ) 2 1) c 1) 1 2) 2 1) c j) j0 2 j 1 1) 1) 2 2)! l0 1) l cj) l1 l! ), l c j) l1 l! ). 83) Corollary 14. i0 i )c i c 1) i c ) i ) 1) c) 1 c) If 2,theU ) 2 ca be epresse eplicitly. Theore 15. For 0a 1oe has ) 0). 87) U 0) 2 ) c, 88) U ) 2 ) c j1 1 2 ) c U ) j0 c j) c j) 1) cj) 1 ), 89) 1) c j) 1 ). 90) Puttig 1i 89), we have the alterative ietity 2.3) i [6, Theore 2.4] because c c c 1 by 2.2) i [6, Theore 2.4]. Corollary 16. Oe has i0 i )c i c i 1) c c 1 ) 1). 91) Rear 12. Note that 1 1 2) 1) 1! l) 1) l c l l!. 84) c 1) 2.4.TwoKisofPoly-CauchyNubers. Defie U ) ) by l0 U ) ) : )c i i 1 i 1,...,i i1 c i 1 1 i 1,...,i 0 c ) i 1, 0). 85) The we obtai the followig. Theore 13. For a iteger a a oegative iteger, oe has l0 1) l [ 1 l1 ]U l) 1 )! { { l0 i0 i! l i ){ l } c) li ) ; { 0 0 1). 86) Puttig 1i Theore 13,wehavethefollowig. If 3,theU ) 3 cabeepresseasfollows. Theore 17. For 0a 1oe has U 0) 3 ) 1) c c 1 ) 1) c, U ) 3 ) U 0) 3 ) 1 2 ) c 1 1) j1 1 2 j 1 ) j) 2) 2 c 2 2 3) cj) 1 cj) ), U ) 3 ) U 0) 3 ) 2 1) c 1 2 1) j0 2 j 1 1) j) 2) 2 c 2 2 3) c j) 1 c j) ). Defie V ) ) by V ) ) : ) i i 1 i 1,...,i c i1 c i 1 1 i 1,...,i 0 Theweobtaithefollowig. c ) i 1, 0). 92) 93) 94) 95)

9 Joural of Discrete Matheatics 9 Theore 18. For a iteger a a oegative iteger, oe has l0 1) l [ 1 l1 ]V l) 1 ) 1) 1 1) i! { l0 i0 i! l i 1 i ){ l }c) li! { )! c) ) ; { 0 0 1). 96) Puttig 1i Theore 18,wehavethefollowig. Corollary 19. i0 i ) c i c 1) i c ) i )c) 0). 97) If 2,theV ) 2 ca be epresse eplicitly. Theore 20. For 0a 1oe has V 0) 2 ) c, 98) V ) c j) j1 2 ) c, 99) 1 2 ) c V ) c j) j0. 100) Puttig 1 i 99), we have the ietity 2.3) i [6, Theore 2.4]. Corollary 21. Oe has i0 i ) c ic i 1) c 1). 101) If 3,theV ) 3 ca be epresse as follows. Theore 22. For 0a 1oe has V 0) 3 ) 1) c, V ) 3 ) V 0) 3 ) 1 2 ) c 1 1) c j) j1 1 2 j 1 ) 1) 2 2)! V ) 3 ) V 0) 3 ) 2 1) c 1 2 1) c j) 3. Further Stuy j0 l0 2 j 1 1) 1) 2 2)! l0 1) 1) l cj) l1 l! ), l c j) l1 l! ). 102) Kaao [14] etioe that eplicit forulae of S ) ) for 4 seee to be coplicate to escribe. We will give eplicit forulae of T ) ) for ay 2later aywhere else. I aitio, oe ay cosier the sus of proucts of ) Cauchy ubers a poly-cauchy ubers. It woul be a iterestig wor to establish the eplicit epressios of such suatios. Acowlegets This wor was supporte i part by the Grat-i-Ai for Scietific research C) o ), the Japa Society for the Prootio of Sciece. Refereces [1] L. Cotet, Avace Cobiatorics, Reiel, Doreecht, The Netherla, [2] T. Agoh a K. Dilcher, Recurrece relatios for örlu ubers a beroulli ubers of the seco i, Fiboacci Quarterly,vol.48,o.1,pp.4 12,2010. [3] P. T. Youg, A 2-aic forula for Beroulli ubers of the seco i a for the Nörlu ubers, Joural of Nuber Theory,vol.128,o.11,pp ,2008. [4] G.S.Cheo,S.G.Hwag,aS.G.Lee, Severalpolyoials associate with the haroic ubers, Discrete Applie Matheatics,vol.155,o.18,pp ,2007. [5] H. M. Liu, S. H. Qi, a S. Y. Dig, Soe recurrece relatios for cauchy ubers of the first i, Joural of Iteger Sequeces,vol.13,o.3,pp.1 7,2010. [6] D. Merlii, R. Sprugoli, a M. C. Verri, The Cauchy ubers, Discrete Matheatics, vol. 306, o. 16, pp , [7] W. Wag, Geeralize higher orer Beroulli uber pairs a geeralize Stirlig uber pairs, Joural of Matheatical Aalysis a Applicatios,vol.364,o.1,pp ,2010. [8]F.Z.Zhao, SusofprouctsofCauchyubers, Discrete Matheatics,vol.309,o.12,pp ,2009.

10 10 Joural of Discrete Matheatics [9] M. Kaeo, Poly-Beroulli ubers, Joural e Théorie es Nobres e Boreau, vol. 9, pp , [10] R.L.Graha,D.E.Kuth,aO.Patashi,Cocrete Matheatics, Aiso-Wesley, Reaig, Mass, USA, 2 eitio, [11] T. Agoh a K. Dilcher, Shortee recurrece relatios for Beroulli ubers, Discrete Matheatics, vol.309,o.4,pp , [12] T. Koatsu, Poly-Cauchy ubers, Kyushu Joural of Matheatics,vol.67,2013. [13] K. Kaao a T. Koatsu, Poly-Cauchy polyoials, I preparatio. [14] K. Kaao, Sus of proucts of Beroulli ubers, icluig poly-beroulli ubers, Joural of Iteger Sequeces, vol. 13, o.5,pp.1 10,2010.

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