Automated Proofs for Some Stirling Number Identities

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1 Autoated Proofs for Soe Stirlig Nuber Idetities Mauel Kauers ad Carste Scheider Research Istitute for Sybolic Coputatio Johaes Kepler Uiversity Altebergerstraße 69 A4040 Liz, Austria Subitted: Sep 1, 2007; Accepted: Dec 4, 2007; Published: Ja 1, 2008 Matheatics Subject Classificatio: 68W30 Abstract We preset coputer-geerated proofs for soe suatio idetities for (-)Stirlig ad (-)Euleria ubers that were obtaied by cobiig a recet suatio algorith for Stirlig uber idetities with a recurrece solver for differece fields. 1 Itroductio I a recet article [5], suatio algoriths for a ew class of seueces defied by certai types of triagular recurrece euatios are give. With these algoriths it is possible to copute recurreces i ad for sus of the for F (, ) = h(,, k)s(, k) k=0 where h(,, k) is a hypergeoetric ter ad S(, k) are, e.g., Stirlig ubers or Euleria ubers. Recall that these ay be defied via S 1 (, k) = S 1 ( 1, k 1) ( 1)S 1 ( 1, k) S 1 (0, k) = δ 0,k, (1) S 2 (, k) = S 2 ( 1, k 1) + ks 2 ( 1, k) S 2 (0, k) = δ 0,k, (2) E 1 (, k) = ( k)e 1 ( 1, k 1) + (k + 1)E 1 ( 1, k) E 1 (0, k) = δ 0,k. (3) kauers@risc.ui-liz.ac.at; Partially supported by FWF grats SFB F1305 ad P16613-N12 cscheid@risc.ui-liz.ac.at; Partially supported by FWF grats SFB F1305. Part of this work was doe while the two authors were attedig the Triestre o ethods of proof theory i atheatics at the Max-Plack-Istitute for Matheatics, Bo, Geray. the electroic joural of cobiatorics 15 (2008), #R2 1

2 The origial algoriths exploit hypergeoetric creative telescopig [9]. More geerally, the algoriths ca be exteded to work for ay seuece h(,, k) that ca be rephrased i a differece field i which oe ca solve creative telescopig probles. Sice such probles ca be solved i Karr s ΠΣ-fields [3, 8], we ca allow for h(,, k) ay idefiitely ested su or product expressio, such as (-)hypergeoetric ters, haroic ubers H k = k i=1, etc. Moreover, S(, k) ay satisfy ay triagular recurrece of the for 1 i S(, k) = a 1 (, k)s( + α, k + β) + a 2 (, k)s( + γ, k + δ) (4) with α, β, γ, δ Z ad α γ β δ = ±1 ad coefficiets a 1 (, k) ad a 2 (, k) that ca be defied by ay idefiite ested su or product over k. I coectio with creative telescopig i ΠΣ-fields, the algoriths of [5] directly exted to this ore geeral class of suads. Give a suad f(,, k) = h(,, k)s(, k) as specified above ad give a fiite set of pairs S Z 2, the algoriths costruct, if possible, expressios c i,j (, ), free of k, ad g(,, k) such that the creative telescopig euatio c i,j (, )f( + i, + j, k) = g(,, k + 1) g(,, k) (5) (i,j) S holds ad ca be idepedetly verified by siple arithetic. Suig (5) over the suatio rage leads to a recurrece relatio, ot ecessarily hoogeeous, of the for c i,j (, )F ( + i, + j) = d(, ). (6) (i,j) S The validity of this recurrece follows, siilar to the hypergeoetric settig [6], fro (5), but is typically ot obvious if (5) is ot available. Therefore, g(,, k) (the oly iforatio cotaied i (5) but ot i (6)) is called the certificate of the recurrece. I the followig sectio, we give a detailed exaple for provig a Stirlig uber idetity ivolvig haroic ubers i this way. A collectio of further idetities about -Stirlig ubers that ca be prove aalogously is give afterwards. 2 A Detailed Exaple Cosider the su F (, ) = ( ) H k ( k)!( 1) k+1 S 1 (k 1, ). k 1 }{{}}{{} =:h(,,k) =:S(,k) }{{} =:f(,,k) Here, S 1 refers to the (siged) Stirlig ubers of the first kid. the electroic joural of cobiatorics 15 (2008), #R2 2

3 The algorith of [5] reduces the recurrece costructio to soe creative telescopig probles which ca be solved by algoriths for ΠΣ fields [7]. The solutios to all these euatios are cobied to the recurrece euatio F (, ) 2F (, + 1) 2F ( + 1, + 1) + 2 F (, + 2) + (2 + 1)F ( + 1, + 2) + F ( + 2, + 2) = S 1 ( 1, + 1) ( 1)S 1 ( 1, + 2), which the algorith returs as output alog with the certificate ( ) (k 1) g(,, k) = (k 3)(k 2) ( 1) k ( k)! k 1 ( (k 2 3k 6k (k 2)(k 1)H k )S 1 (k 1, + 2) + (k 3)((k 1)H k 1)S 1 (k 1, + 1) ). The certificate g(,, k) allows us to verify the recurrece for F (, ) idepedetly. Ideed, usig the triagular recurrece (1) for S 1 ad the obvious relatios for factorials, haroic ubers, etc. it is readily checked that f(,, k) 2f(, + 1, k) 2f( + 1, + 1, k) + 2 f(, + 2, k) + (2 + 1)f( + 1, + 2, k) + f( + 2, + 2, k) = g(,, k + 1) g(,, k). Now su this euatio for k = 1,..., 1. This gives 1 1 f(,, k) f(, + 1, k) 2 1 f(, + 2, k) + (2 + 1) ( ) = g(,, k + 1) g(,, k). f( + 1, + 1, k) f( + 1, + 2, k) + 1 f( + 2, + 2, k) The right had side collapses to g(,, ) g(,, 1). O the left had side, we ca express the sus i ters of the F ( + i, + j) usig, e.g., 1 f( + 1, + 2, k) = F ( + 1, + 2) f( + 1, + 2, ) f( + 1, + 2, + 2). Brigig fially everythig but the F ( + i, + j) to the right had side ad doig soe straightforward siplificatios gives the recurrece claied by the algorith. the electroic joural of cobiatorics 15 (2008), #R2 3

4 With the recurrece for F (, ) at had, it is a easy atter to prove the closed for represetatio F (, ) = 1 2 ( + 1)( + 2)S 1(, + 2). Just check that the closed for satisfies the sae recurrece (this is easy) ad a suitable set of iitial values. The creative telescopig probles arisig durig the executio of the algorith are iterestig also fro a coputatioal poit of view. Oe of these euatios, as a exaple, is (k 1)(k 1)((k )H k +1) k(k ) 2 H k b 2 (,, k + 1) b 2 (,, k) c 2,0 (, ) + (+1)(( k+1)h k+1) ( k+2)h k c 2,1 (, ) (+1)(+2)(( k+1)h k +1)(( k+2)h +1 k +1) ( k+2)( k+3)h k H +1 k c 2,2 (, ) = 0, where b 2 (,, k) ad the c i (, ) are to be deteried. This euatio differs fro ost euatios arisig fro atural (o-stirlig-) sus i that haroic uber expressios also arise i deoiators. 3 Soe -Idetities Subseuetly, we cosider soe -versios of the well-kow idetities ( ) S 2 (k, ) = S 2 ( + 1, + 1), (7) k k= ( ) k ( 1) k S 1 (, k) = ( 1) S 1 ( + 1, + 1). (8) k= Followig Gould [2], we defie the -Stirlig ubers via S () 1 (, k) = 1 S () 1 ( 1, k 1) [ 1]S () 1 ( 1, k), S () 1 (0, k) = δ 0,k, S () 2 (, k) = k 1 S () 2 ( 1, k 1) + [k]s() 2 ( 1, k), S() 2 (0, k) = δ 0,k, where [] = ( 1)/( 1) ad δ refers to the Kroecker delta. By [ ] we deote the k -bioial coefficiet, defied as [ ] = []!/[k]!/[ k]!. k 1. We prove the idetity [4, Id. 1] ( ) k S () 2 (k, ) = S() 2 ( + 1, + 1) k k= by coputig the recurrece (1 )F ( + 1, + 1) (1 ) +2 F (, ) (1 +2 )F ( + 1, ) = 0 the electroic joural of cobiatorics 15 (2008), #R2 4

5 for the su F (, ) = k= k( ) () k S 2 (k, ) ( ) k( 1)k+1 g(,, k) = S () 2 (k, + 1). k 1 k 2. The idetity [4, Id. 2] ( ) k ( 1) k k= follows fro the recurrece S () 1 (, k) k = ( 1) S () 1 ( + 1, + 1) ( 1) +1 F ( + 1, + 1) + ( 1)F (, ) + ( +1 1)F ( + 1, ) = 0 3. For the su g(,, k) = ( 1) k ( k)( 1) 1 k + 1 F (, ) = k= ( ) k ( 1) k S 1 (, k) k, ivolvig a -bioial, we copute the recurrece relatio S () 1 (, k 1). F (, ) + ( + )F ( + 1, ) F ( + 1, + 1) = 0 g(,, k) = ( 1) k ( k ) +k ( +1 1) S 1 (, k 1). This yields aother -versio of idetity (8). Naely, defie S () 1 (, k) by S () 1 ( + 1, k + 1) = 1 S() 1 (, k) (k + ) () S 1 (, k + 1) () ad S 1 (0, k) = δ 0,k. Observe that i the liit 1 this also specializes to S 1 (, k). The by costructio we get the -versio ( 1) k S 1 (, k) k = ( 1) k= S() 1 ( + 1, + 1). 4. For F (, ) = ( 1) k S () 1 (, k) k k= the electroic joural of cobiatorics 15 (2008), #R2 5

6 we copute the recurrece ( 1) +1 F ( + 1, + 1) + ( )F ( + 1, ) + ( 1)F (, ) = 0 If we defie g(,, k) = ( 1) k ( 1)( k ) +k ( +1 1) S () 1 (, ) by S () 1 ( + 1, k + 1) = 1 (1 ) ( k + k+1 + 1) S () 1 (, k 1). S () 1 ( + 1, k) + 1 S() 1 (, k) () ad S 1 (0, k) = δ 0,k, which specializes i the liit 1 to S 1 (, k), we arrive at the -versio ( 1) k S 1 (, k) k = ( 1) S() 1 ( + 1, + 1). k= 5. Carlitz [1] defies the -Euleria ubers E () 1 (, ) by reuestig that they satisfy +1 [] = E () 1 (, k) [ ] + k 1 which is a -aalogue of the Worpitzky idetity [1]. He derives the recurrece euatio E () 1 ( + 1, k) = [ + 2 k]e () 1 (, k 1) + +1 k [k]e () 1 (, k). Coversely, takig this recurrece euatio ad suitable iitial coditios as the defiitio of the -Euleria ubers, we fid that the su satisfies the recurrece the certificate beig +1 [ ] + k 1 F (, ) = E () 1 (, k) ( 1)F (, ) ( 1)F ( + 1, ) = 0, g(,, k) = k 1 ( k+ )( k +2 ) +1 1 The idetity F (, ) = [] follows easily., + 2 E () 1 (, k 1). the electroic joural of cobiatorics 15 (2008), #R2 6

7 Reark. A closed for represetatio caot be foud for every su, but alost always it is possible to costruct a recurrece euatio. For istace, for F (, ) = we copute the recurrece relatio k( 1) k S 1 (, k) k k= 2 ( +1 + ) 2 F ( + 2, ) + 2 ( )F ( + 2, + 1) 2 F ( + 2, + 2) ( )F ( + 1, ) + 2F ( + 1, + 1) F (, ) = 0 g(,, k) = ( 1) k k 2+1 ( k )[ k ] ((k 1)( k+1 1)S 1 (,k 1) +k( k +1 )S 1 (,k 2)) Refereces [1] L. Carlitz. -Beroulli ad Euleria ubers. Trasactios of the AMS, 76(2): , [2] H. W. Gould. The -Stirlig ubers of first ad secod kid. Duke Joural of Matheatics, 28(2): , [3] M. Karr. Suatio i fiite ters. J. ACM, 28: , [4] J. Katriel. Stirlig uber idetities: itercosistey of aalogues. J. Phys. A: Math. Ge., 31: , [5] M. Kauers. Suatio Algoriths for Stirlig Nuber Idetities. Joural of Sybolic Coputatio, 42(11): , [6] M. Petkovšek, H. Wilf, ad D. Zeilberger. A = B. AK Peters, Ltd., [7] C. Scheider. Solvig paraeterized liear differece euatios i ters of idefiite ested sus ad products. J. Differ. Euatios Appl., 11(9): , [8] C. Scheider. Sybolic suatio assists cobiatorics. Sé. Lothar. Cobi., 56:1 36, Article B56b. [9] D. Zeilberger. The ethod of creative telescopig. J. Sybolic Coput., 11: , the electroic joural of cobiatorics 15 (2008), #R2 7