Expression for Restricted Partition Function through Bernoulli Polynomials

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1 Expressio for Restricted Partitio Fuctio through Beroulli Polyomials Boris Y. Rubistei Departmet of Mathematics, Uiversity of Califoria, Davis, Oe Shields Dr., Davis, CA 9566, U.S.A. February 28, 2005 Abstract Explicit expressios for restricted partitio fuctio W s, d m ad its quasiperiodic compoets W s, d m called Sylvester waves for a set of positive itegers d m = {d, d 2,..., d m } are derived. The formulas are represeted i a form of a fiite sum over Beroulli polyomials of higher order with periodic coefficiets. Itroductio The problem of partitios of positive itegers has log history started from the wor of Euler who laid a foudatio of the theory of partitios [], itroducig the idea of geeratig fuctios. May promiet mathematicias cotributed to the developmet of the theory usig this approach. J.J. Sylvester provided a ew isight ad made a remarable progress i this field. He foud [, 2] the procedure for computatio of a restricted partitio fuctio, which he called deumerat or quotity, ad described symmetry properties of such fuctios. The restricted partitio fuctio W s, d m W s, {d, d 2,..., d m } is a umber of partitios of s ito positive itegers {d, d 2,..., d m }, each ot greater tha s. The geeratig fuctio for W s, d m has a form F t, d m = m t d = W s, d m t s, i where W s, d m satisfies the basic recursive relatio s=0 W s, d m W s d m, d m = W s, d m. Sylvester also proved the statemet about splittig of the partitio fuctio ito periodic ad operiodic parts ad showed that the restricted partitio fuctio may be preseted as a sum of waves, which we call the Sylvester waves W s, d m = = W s, d m, 2 where summatio rus over all distict factors of the elemets i the set d m. The wave W s, d m is a quasipolyomial i s closely related to prime roots ρ of uity. Namely, Sylvester showed i [2] that the wave W s, d m is a coefficiet of t i the series expasio i ascedig powers of t of F s, t = ρ ρ e st m = ρ d. 3 e d t

2 The summatio is made over all prime roots of uity ρ = exp2πi/ for relatively prime to icludig uity ad smaller tha. This result is ust a recipe for calculatio of the partitio fuctio ad it does ot provide explicit formula. Usig the Sylvester recipe we foud i [0] a explicit formula for the Sylvester wave W s, d m i a form of fiite sum of the Beroulli polyomials of higher order [2, 8] multiplied by a periodic fuctio of iteger period. The periodic factor is expressed through the Euleria polyomials of higher order [4]. I this ote we show that it is possible to express the partitio fuctio through the Beroulli polyomials oly. A special symbolic techique is developed i the theory of polyomials of higher order, which sigificatly simplifies computatios performed with these polyomials. A short descriptio of this techique required for better uderstadig of this paper is give i the Appedix. 2 Polyomial part of partitio fuctio ad Beroulli polyomials Cosider a polyomial part of the partitio fuctio correspodig to the wave W s, d m. It may be foud as a residue of the geerator F s, t = e st e d it. 4 Recallig the geeratig fuctio for the Beroulli polyomials of higher order [2]: e st t m d i edit = B m s d m t!, ad a trasformatio rule B m s d m = B m s + d i d m, we obtai the relatio e st e d it = π m B m s + s m d m t m!, 5 where s m = m d i, π m = d i. It is immediately see from 5 that the coefficiet of /t i 4 is give by the term with = m W s, d m = B m m m! π s + s m d m. 6 m The polyomial part also admits a symbolic form frequetly used i theory of higher order polyomials m W s, d m = s + s m + d i B i, m! π m 2

3 where after expasio powers r i of i B are coverted ito orders of the Beroulli umbers i B r i B ri. It is easy to recogize i 6 the explicit formula reported recetly i [3], which was obtaied by a straightforward computatio of the complex residue of the geerator 4. Note that basic recursive relatio for the Beroulli polyomials [8] B m s + d m d m B m s d m = d m B m s d m aturally leads to the basic recursive relatio for the polyomial part of the partitio fuctio: W s, d m W s d m, d m = W s, d m, which coicides with. This idicates that the Beroulli polyomials of higher order represet a atural basis for expasio of the partitio fuctio ad its waves. 3 Sylvester waves ad Euleria polyomials Frobeius [7] studied i great detail the so-called Euleria polyomials H s, ρ satisfyig the geeratig fuctio ρe st e t = H s, ρ t, ρ, ρ! which reduces to defiitio of the Euler polyomials at fixed value of the parameter ρ E s = H s,. The polyomials H ρ H 0, ρ satisfy the symbolic recursio H 0 ρ = The geeralizatio to higher orders is straightforward e st ρd i ed it ρ d i = ρh ρ = Hρ +, > 0. 7 where the correspodig recursive relatio for H m H m s + d m, ρ d m ρ dm H m s, ρ d m = H m s, ρ d m t!, ρdi, 8 s, ρ d m has the form ρ dm H m s, ρ d m. The Euleria polyomials of higher order H m s, ρ d m itroduced by L. Carlitz i [4] ca be defied through the symbolic formula H m s, ρ d m = where H ρ computed from the relatio s + d i i Hρ d i, ρ e t ρ = H ρ t!, or usig the recursio 7. Usig the polyomials H m s, ρ d m we ca compute the Sylvester wave of arbitrary period. 3

4 I order to compute the Sylvester wave with period > we ote that the summad i the expressio 3 ca be rewritte as a product F s, t = ρ e st e dit ρ i= + ρd i e dit, 9 where the elemets i d m are sorted i a way that is a divisor for first elemets we say that has weight, ad the rest elemets i the set are ot divisible by. Cosider a -periodic Sylvester wave W s, d m, ad rewrite the summad i 9 as double ifiite sum usig 5 ad 8 π ρ i= + ρd i B s + s d t! l=0 H m l s m s, ρ d m tl l!, 0 The coefficiet of /t i 0 is foud for + l =, so that we obtai a fiite sum: W s, d m = ρ! π m i= + ρd i This expressio may be rewritte as a symbolic power : W s, d m ρ =! π m s i= + ρd i + s m + ρ ρ B s + s d H m s m s, ρ d m. d i i B + d i i Hρ d i i= +, which is equal to W s, d m =! π ρ i= + ρd i ρ Hm B s + s m d [ρ d m ], 2 where H m [ρ d m ] = H m 0, ρ d m = are Euleria umbers of higher order ad it is assumed that [ m d i i Hρ d i ], H 0 0 [ρ ] =, H0 [ρ ] = 0, > 0. It should be uderlied that the presetatio of the Sylvester wave as a fiite sum of the Beroulli polyomials with periodic coefficiets 2 is ot uique. The symbolic formula ca be cast ito a sum of the Euleria polyomials W s, d m =! π ρ i= + ρd i ρ 4 Hm B [d ] s + s m, ρ d m, 3

5 where B m [d m ] = B m 0 d m are the Beroulli umbers of higher order. It should be oted that the formulae 2 ad 3 require summatio over all prime roots of uity, ad though it is simpler tha the Sylvester recipe usig 3, it caot be cosidered a completely explicit formula. 4 Reductio of Sylvester waves to Beroulli polyomials A relatio betwee the Euleria ad Beroulli umbers ad polyomials of higher order established i [4] may be writte as follows: m! π m ρ sm! π ρ s m = m i= + r i =0 i= + ρd i ρ d ir i Usig this relatio we covert the ier sum i 2 ito ρ [ρ d m ] Hm B m m m ρ Hm m i= + ρd i [ρ d m ] i= + r i =0 ρ d ir i i= + i= + d i r i d m. = m! π ρ s m m! π m ρ m B m m m d i r i d m = m! π m! π m B m m m i= + m d i r i d m Ψ s + i= + r i =0 i= + Here Ψ s deotes a prime radical circulator itroduced i [5] see also [6] d i r i +. 4 Ψ s = ρ ρ s. For prime it is give by the Ψ s = { φ, s = 0 mod, µ, s 0 mod. where φ ad µ deote Euler totiet ad Möbius fuctios. powers of distict prime factors = p α, 5 Cosiderig as a product of 5

6 oe may easily chec that for iteger values of s Ψ s = p α Ψ p s p α, 6 where Ψ s = 0 for o-iteger values of s. It is coveiet to itroduce a -modified set of summads d m defied as uio of subset d of summads divisible by ad the remaiig part d m multiplied by d m = d d m, so that d m is divisible by. Substitutio of 4 ito 2 with extesio of the outer summatio up to m produces the formula for computatio of the -periodic Sylvester wave: W s, d m m = m! π m m B m m s + s m + i= + i= + r i =0 d i r i d m Ψ s + i= + d i r i +. 7 The derivatio of 7 implies that all terms cotaiig s i power larger tha idetically equal to zero. The polyomial part of the partitio fuctio W s, d m for the -modified set of summads reads: W s, d m = m! π m m Bm m s + s m + i= + d i d m. This formula gives rise to the represetatio of the -periodic Sylvester wave W s, d m through the liear combiatio of the polyomial part of the -modified set of summad d m multiplied by the -periodic fuctios Ψ : m W s, d m = W s + d i r i, d m Ψ s + d i r i, which is writte also as i= + r i = W s, d m = m i= + r i =0 i= + i= + i= + W s d i r i, d m Ψ s d i r i. 8 i= + The last formula shows that each Sylvester wave is expressed as a liear superpositio of the polyomial parts of the modified set of summads multiplied by the correspodig prime circulator. Thus, the formulae 5,6,8 with the Sylvester splittig formula 2 provide the explicit solutio of the restricted partitios problem. Appedix The symbolic techique for maipulatig sums with biomial coefficiets by expadig polyomials ad the replacig powers by subscripts was developed i ieteeth cetury by Blissard. It has 6

7 bee ow as symbolic otatio ad the classical umbral calculus [9]. A example of this otatio is also foud i [2] i sectio devoted to the Beroulli polyomials B x. The well-ow formulas B x + y = are writte symbolically as =0 B xy, B x = =0 B x + y = Bx + y, B x = B + x. B x, After the expasio the expoets of Bx ad B are coverted ito the orders of the Beroulli polyomial ad the Beroulli umber, respectively: [Bx] B x, B B. We use this otatio i its exteded versio suggested i [8] i order to mae derivatio more clear ad itelligible. Nörlud itroduced the Beroulli polyomials of higher order defied through the recursio B m x d m = =0 d mb B m x d m, startig from B x d = d B x d. I symbolic otatio it taes form ad recursively reduces to more symmetric form B m x = d m B + B x m, B m x d m = x + d B + d 2 2 B d m m B = x + where each [ i B] is coverted ito B. Refereces [] G. E. Adrews, The Theory of Partitios, Ecyclopedia of Mathematics ad its Applicatios, V.2, Addiso Wesley, 976. [2] H. Batema ad A. Erdelýi, Higher Trascedetal Fuctios, V., McGraw-Hill Boo Co., NY, 953. d i i B, [3] M. Bec, I. M. Gessel ad T. Komatsu, The Polyomial Part of a Restrictio Partitio Fuctio Related to the Frobeius Problem, The Electroic Joural of Combiatorics, 8, N7 200, -5. [4] L. Carlitz, Euleria Numbers ad Polyomials of Higher Order, Due Mathematical Joural, , [5] A. Cayley, Researches o the Partitios of Numbers, Phyl. Tras. Royal Soc ,

8 [6] L. Dicso, History of the Theory of Numbers, V.2, Ch.3, Chelsea Publishig Compay, NY, 976. L. Comtet, Advaced Combiatorics, Ch.2, D. Reidel Publishig Co., Dordrecht, Hollad, 974. [7] F. G. Frobeius, Über die Beroullische Zahle ud die Eulerische Polyome, Sitzugsberichte der Köiglich Preußische Aademie der Wisseschafte zu Berli 90, [8] N. E. Nörlud, Mémoire sur les Polyomes de Beroulli, Acta Mathematica, , [9] S. Roma ad G.-C. Rota, The Umbral Calculus, Adv. Math , [0] B.Y. Rubistei ad L.F. Fel, Restricted Partitio Fuctio as Beroulli ad Euler Polyomials of Higher Order, Ramaua Joural, accepted. [] J. J. Sylvester, O the Partitio of Numbers, Quarterly Joural of Mathematics 857, [2] J. J. Sylvester, O Subivariats, i.e. Semi-ivariats to Biary Quatics of a Ulimited Order. With a Excursus o Ratioal Fractios ad Partitios, America Joural of Mathematics 5 882,