3 47 6 3 Jounl of Intege Sequences, Vol. 3 (, Aticle..8 Integls nd Polygmm Repesenttions fo Binomil Sums Anthony Sofo School of Engineeing nd Science Victoi Univesity PO Box 448 Melboune City, VIC 8 Austli nthony.sofo@vu.edu.u Abstct We conside sums involving the poduct of ecipocl binomil coefficiend polynomil tems nd develop some double integl identities. In pticul cses it is possible to expess the sums in closed fom, give some genel esults, ecove some known esults in Coffey nd poduce new identities. Intoduction In ecent ppe Coffey [8] consides summtions ove digmm nd polygmm functions nd develops mny esults, nmely two of his popositions, in tems of the Riemnn zet function ζ (, e espectively equtions (59 nd (66 nd n ( n n p+ (n + ( p ( ln + n n p+ (n + ( p + p ( p+m ( m ζ (m + ( m p ( p+m ζ (m +. ( Coffey [8] lso constucts new integl epesenttions fo these sums. The mjo im of this ppe is to investigte genel binomil sums with vious pmetes tht then enbles one to give moe genel epesenttions of ( nd (, theeby genelizing the popositions of m
Coffey, both in closed fom in tems of zet functions nd digmm functions t possible tionl vlues of the gument, nd in double integl fom. The following definitions will be useful. The Psi, o digmm function ψ (z, is defined by ψ (z d dz log Γ (z Γ (z Γ (z ( n + γ n + z whee γ denotes the Eule-Mscheoni constnd Γ (z is the Gmm function. Similly nd ψ (z + n n ( n γ, (3 n + z ( ψ (z ψ (z + ψ z + + ln. (4 Sums of ecipocls of binomil coefficients ppe in the clcultion of mssive Feynmn digms [3] within sevel diffeent ppoches: fo instnce, s solutions of diffeentil equtions fo Feynmn mplitudes, though nive ε-expnsion of hypegeometic functions within Mellin-Bnes technique o in the fmewok of ecently poposed lgebic ppoch []. Thee hs ecently been enewed inteest in the study of seies involving binomil coefficients nd numbe of uthos hve obtined eithe closed fom epesenttion o integl epesenttion fo some pticul cses of these seies. The inteested ede is efeed to [,, 3, 4, 5, 6, 9,, 6, 7, 8, 9,,,, 3, 5, 6, 7]. The min esults The following Lemm nd well-known definition will be useful in the poof of the min theoem. Definition. Let z, m nd q,, 3,... Then z (q + m ( m (m! The next Lemm dels with two infinite sums. Lemm. Let nd be positive el numbes. Then n n (n + H n ( n n (n + { z y q (ln (y m dy. (5 zy ( H ( nd (6 } H (. (7
Poof. n n (n + ( n n + n [ ( ] γ + ψ + nd fom (3 H( ; hence (6 is ttined. n ( n n (n + [ n [ γ + ψ ( n ( + [ ( ψ + ψ ( n + n n + n n ( + γ ψ + ] ln ( + ] ln, ] n (n fom the definition (4 ( ψ + ( ( ψ + ψ + ln ; hence n ( n n (n + [ ( ψ [ H ( + γ H ( ( ] ψ + ] + γ, theefoe (7 follows. Remk 3. In the following Coollies nd emks we encounte hmonic numbes t possible tionl vlues of the gument, of the fom H (α whee,, 3,...,k, α,, 3,... nd k N. The polygmm function ψ (α (z is defined s: ψ (α (z dα+ dα [log Γ (z] [ψ (z], z {,,, 3,...}. dzα+ dzα To evlute H (α we hve vilble eltion in tems of the polygmm function ψ (α (z, fo tionl guments z, we lso define H (α+ H ( ( ζ (α + + ( α ψ (α α! + ( γ + ψ +, nd H (α. 3
The evlution of the polygmm function ψ ( (α t tionl vlues of the gument cn be explicitly done vi fomul s given by Kölbig [5], (see lso [4], o Choi nd Cvijovic [7] in tems of the polylogithmic o othe specil functions. Some specific vlues e given s, mny othes e listed in the book [4]: ( ψ (n ( n n! ( n+ ζ (n + H ( 3 H ( 4 4 π 3 π 3 3 ln 3 We now stte the following theoem. 3 ln (, H( 3 4, nd H( 5 6 4 3 + π 3 ln (, 6 5 + 3π 3 ln (3 ln (. Theoem 4. Let be positive el numbe, t, j, nd k N {}. Then S k+ (, j, t (j+t( k k! n n k+ ( n + j + j + ( x j x (ln(y k tx y dxdy, fo k ( x t j+ x dx, fo k tx (k+ tems {}}{,,...,, T +k+ F +k,,...,, + j + }{{} k tems whee pf q [ ] is the genelized hypegeometic function, nd B (, is the bet function. Poof. Conside ( tems {}}{ +,...,,..., + j + + } {{ } tems T t (j + B (j +, +, (8. t (9 n n k+ ( n + j + j + n (j + (j + Γ (j + Γ (n + n k+ Γ (n + j + n B (n +,j + nk+ 4
now eplcing the bet function with its integl epesenttion, we hve (j + n n B (n +,j + (j + k+ n k+ By justified chnging the ode of integtion nd summtion we hve, n x n ( x j dx. n n k+ ( n + j + j + (j + ( x j (tx n n n k+ dx (j + t( k k! ( x j x (ln (y k dxdy, fo k tx y upon utilizing Definition. The cse of k follows in simil wy so tht t S (, j, t ( n n n + j + n j + t ( x j+ x tx dx; hence the integls in (8 e ttined. By the considetion of the tio of successive tems whee U n+ U n we obtin the esult (9. U n n k+ ( n + j + j + The following inteesting coollies follow fom Theoem 4. Coolly 5. Let t nd >. Also let j nd k be integes. Then S k+ (, j, ( n n + j + n k+ j + (j + ( k k! ( x j x (ln (y k dxdy, fo k ( x y k j+ A s (j +!ζ (k + s + s ( +k+ ( ( k j + H ( ( 5
whee Poof. By expnsion, [ { }] d s n k A s lim n s! dn s n k j+, s,,,...k. ( (n + n n k+ ( n + j + j + n n (j +! n k+ (n + j+ (j +! n [ k s (j +! n k+ j+ (n + n j+ A s n + k s B n + ], whee { } n + B lim j+ n ( (n + ( +k+ (j +! A s is defined by (. Hence, fte intechnging the sums, we hve ( n + j + n k+ j + n (j +! [ k j+ A s n + k+ s s n j+ k (j +! A s ζ (k + s + s B n ( ] n (n + ( +k+ ( ( k j +, k ( j + H ( upon utilizing Lemm, which is the esult (. The degenete cse, fo j, gives the known esult ζ (k +. nk+ n The integl ( follows fom the integl in (8. A simil esult is evident fo the cse t. 6
Coolly 6. Let t nd >. Also let j nd k be integes. Then ( n S k+ (, j, ( n n + j + n k+ j + (j + ( k+ k! ( x j x (ln (y k dxdy, fo k + x y k A s (j +! ( s k ζ (k + s (3 s j+ + ( +k+ ( k ( j + ( H ( H ( Poof. The poof, uses (7 nd follows the sme detils s tht of Coolly 5, nd will not be given hee. The ddition nd subtction of ( nd (3 gives us the following epesenttions. Remk 7. nd k n Let > nd let j nd k be integes. Then n k+ ( n + j + j + k j+ A s (j +! s k ζ (k + s + s n (n k+ ( n + j + j + ( +k+ ( k A s (j +! ( s k j+ ζ (k + s + s k ( j + ( +k+ ( ( H ( We give the following exmple to illustte some of the bove identities. Exmple 8. Let k 4, fom ( A A 3 (j +!, A H( j+ (j +!, A 3 ( + 3H ( ( 3 H ( j+ 6 (j +! j+ H( j+ + H(3 j+ 7 ( k j + ( ( H ( ( j+ + H j+ (j +! H ( H (.
theefoe nd n n Remk 9. n 5 ( n + j + j + ( n n 5 ( n + j + j + (j +! [A ζ (5 + A ζ (4 + A ζ (3 + A 3 ζ (] j+ + ( + ( ( 4 j + H ( [ (j +! 5A 6 ζ (5 7A 8 ζ (4 3A 4 ζ (3 A ] 3 ζ ( j+ + ( ( 4 ( j + ( H ( H (. The vey specil cse of nd j llows one to evlute [ { }] d s n k A s lim, s,,,...k n s! dn s n k (n + ( s, nd fom ( nd (3 we cn esily obtin ( nd (. A ecuence eltion fo degenete cse, j, of Theoem 4 is embodied in the following coolly. Coolly. Let the conditions of Theoem 4 hold with j nd put Sk+ : Sk+ (,t n k+ (n + n (k+ tems {}}{ t,,...,, + k+3f k+ +,,...,, }{{} + t, (k+ tems then with solution S k+ + S k Li k+ (t, fo k S k+ ( k S + ( k S + k ( t Φ (t,k +, k ( Li k+ (t 8
whee S (,t n n (n + t + [,, + 3F, + ] t nd Φ, Li e the Lech tnscendend the polylogithm espectively. Poof. We notice tht S k+ + S k nd hence the solution follows by itetion. n n k+ Li k+ (t, Relted esults my be seen in Coffey [, Lemms nd ]. Some exmples e s follows: Fo t, we know tht Li k+ ( ζ (k +. Hence S k+ (, ( k S (, + k ( ζ (k +, fo k. When, we obtin Coffey s [8] esult, by noting tht, fom (6, S (, When, S k+ (, ( k + S k+ k ( ζ (k +. (, 3 ( k + k+ k ( ζ (k +. Similly Sk+ (8, ( k 3(k+ ( k 3k π 3 + ( k 3k+ ln ( ( ( k 3k 3/ ln 3 + k + ( 3( ζ (k +. Fo t, Li k+ ( η (k + ( k ζ (k +, whee η ( is the Diichlet Et function. Hence k Sk+ (, ( k S (, + ( ( k ζ (k +, fo k. When, we obtin Coffey s [8] esult, by noting tht k Sk+ (, ( k ( ln + ( ( k ζ (k +. 9
When 4, ( Sk+ (4, ( k 4 ( ln + π ln + k ( ( ( k ζ (k +, nd similly S k+ ( 8, ( k 533 k ( ( 84 3k + k ζ (k + 8 (k+ tems {}}{ 8,,...,, 9 k+3f k+ 9,,...,,. }{{} (k+ tems Refeences [] H. Alze, D. Kynnkis nd H. M. Sivstv. Seies epesenttions fo some mthemticl constnts. J. Mth. Anl. Appl. 3 (6, 45 6. [] H. Alze nd S. Koumndos. Seies nd poduct epesenttions fo some mthemticl constnts. Peiod. Mth. Hung.58(, (9, 7 8. [3] N. Bti. Integl epesenttions of some seies involving ( k k k n nd some elted seies. Appl. Mth. Comp.47 (4, 645 667. [4] N. Bti. On the seies k ( 3k k k n x k. Poceedings of the Indin Acdemy of Sciences: Mthemticl Sciences, 5(4 (5, 37 38. [5] J. M. Bowein nd R. Gigensohn. Evlutions of binomil seies. Aequtiones Mth., 7, (5, 5 36. [6] J. M. Bowein, D. J. Bodhusd J. Kmnitze. Centl binomil sums, multiple Clusen vlues nd zet vlues. Expeiment. Mth.,, (, 5 34. [7] J. Choi nd D. Cvijovic, Vlues of the polygmm functions t tionl guments, J. Phys. A: Mth. Theo. 4 (7, 59 58. [8] M. W. Coffey. On one dimensionl digmm nd polygmm seies elted to the evlution of Feynmn digms. J. Comput. Appl. Mths., 83, (5, 84. [9] M. W. Coffey. On some seies epesenttions of the Huwitz zet function. J. Comput. Appl. Mths. 6, (8, 97- -35.
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[7] I. J. Zucke. On the seies k ( k k k n nd elted sums. J. Numbe Theoy,, (985, 9. Mthemtics Subject Clssifiction: Pimy 5A; Secondy B65, 5A9, 33C. Keywods: double integl, combintoil identity, hmonic numbe, polygmm function, ecuence. Received Decembe 7 9; evised vesion eceived Febuy 6. Published in Jounl of Intege Sequences, Febuy 3. Retun to Jounl of Intege Sequences home pge.