REVITALIZED AUTOMATIC PROOFS: DEMONSTRATIONS. 1. Introduction., for 1 k n, and the all-familiar Catalan numbers C n =

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1 REVITALIZED AUTOMATIC PROOFS: DEMONSTRATIONS TEWODROS AMDEBERHAN, DAVID CALLAN, HIDEYUKI OHTSUKA AND ROBERTO TAURASO Abstract We cosider three problems from the recet issues of the America Mathematical Mothly ivolvig differet versios of Catala triagle Our mai results offer geeralizatios of these idetities ad demostrate automated proofs with additioal twists, ad o occasio we furish a combiatorial proof 1 Itroductio Let s fix some omeclature The set of all itegers ( is Z, ad the set of o-egative itegers is N Deote the Catala triagle by B, ( +, for 1, ad the all-familiar Catala umbers C +1( 1 correspod to B,1 O the other had, t, ( ( 1 form yet aother variatio of the Catala triagle ad these umbers cout lattice paths (N ad E uit steps from (0, 0 to (, that may touch but stay below the lie y x Covetio Empty sums ad empty products are evaluated to 0 ad 1, respectively Also that ( 0 wheever < 0 or > Let Q a,b : ( a+b a Whe cosiderig a triple product of the umbers B,, o occasio we fid the followig as a more hady reformulatio ( ( ( abc Q a,b Q b,c Q c,a a + b b + c c + a (11 B a, B b, B c, 3 Q a,a Q b,b Q c,c a + b + c + The impetus for this paper comes from Problem [1], Problem [2] ad Problem [3] of the America Mathematical Mothly joural, plus the followig idetities that came up i our study: (12 (13 ( + m ( 2 ( 2m ( 2m + m + 2 m + ( ( ( + m 2m m + m ( ( ( 2m m ( + j m 1 m 1 ( + j, 1 ( ( + j m + j m 1 The purpose of our wor here is to preset certai geeralizatios ad to provide automatic proofs as well as alterative techiues Our demostratio of the Wilf-Zeilberger style of proof [8] exhibit the power of this methodology, especially where we supplemeted it with ovel adjustmets wheever a direct implemetatio ligers Date: July 8,

2 2 TEWODROS AMDEBERHAN, DAVID CALLAN, HIDEYUKI OHTSUKA AND ROBERTO TAURASO A class of d-fold biomial sums of the type R( : 1,, d i1 d ( f( 1,, d + i have bee ivestigated by several authors, see for example [4] ad refereces therei Oe iterpretatio is this: 4 d R( is the expectatio of f if oe starts at the origi ad taes radom steps ± 1 i each of the d dimesios, thus arrivig at the poit ( 2 1,, d Z d with probability d ( 4 d + i i1 The orgaizatio of the paper is as follows I Sectio 2, Problems 11844, ad some geeralized idetities are proved Sectio 3 resolves Problem ad highlights a combiatorial proof together with -aalogue of related idetities Fially, i Sectio 4, we coclude with further geeralizatios ad some ope problems for the reader 2 The first set of mai results Our first result proves Problem of the Mothly [1] as metioed i the Itroductio Lemma 21 For o-egative itegers m, we have (21 ( 3 ( m m (m 2 (m m 1 ( + j ( + j m 1 Proof We apply the method of Wilf-Zeilberger [8] This techiues wors, i the preset case, after multiplyig (21 through with ( 1 m Deote the resultig summad o the LHS of (21 by F 1 (m, ad its sum by f 1 (m : F 1(m, Now, itroduce the compaio fuctio (2m G 1 (m, : F 1 (m, (m 2(m ad chec that F 1 (m + 1, F 1 (m, G 1 (m, + 1 G 1 (m, Telescopig gives f 1 (m + 1 f 1 (m F 1 (m + 1, F 1 (m, [G 1 (m, + 1 G 1 (m, ] G 1 (m, ( 1 m+1 ( m 3 (2m + 1 Let F 2 (m, j be the summad o the RHS of (21 ad its sum f 2 (m : m 1 F 2 (m, j Itroduce j(m j 1 G 2 (m, j : F 2 (m, j (m 2

3 ad chec that F 2 (m + 1, j F 2 (m, j G 2 (m, j + 1 G 2 (m, j Summig 0 j m ad telescopig, we arrive at f 2 (m + 1 f 2 (m m F 2 (m + 1, j m F 2 (m, j + F 2 (m, m m [G 2 (m, j + 1 G 2 (m, j] + F 2 (m, m G 2 (m, m F 2 (m, m ( 1 m+1 ( m 3 (2m The fial step is settled with f 1 (0 f 2 (0 0 (if m 0, so is 0 Theorem 22 For oegative itegers r, s ad m, we have (m 2 ( ( m+r+s m ( m+2r ( m+2s m +r 1 ( m,r,s +r +s + j (22 (m ( m+2r m+r ( m+2s m+s ( m+s +s ( + j + s m + s 1 Proof Agai we use the W-Z method Multiply through euatio (22 by ( m+s +s ad deote the summad o the ew LHS of (22 by F 1 (r, ad its sum by f 1 (r : F 1(r, Now, itroduce the compaio fuctio G 1 (r, : F 1 (r, (s + (m 2(m + r + 1 ad (routiely chec that F 1 (r + 1, F 1 (r, G 1 (r, + 1 G 1 (r, Telescopig gives f 1 (r + 1 f 1 (r F 1 (r + 1, F 1 (r, [G 1 (r, + 1 G 1 (r, ] ( ( ( m + s m + r m + r + s G 1 (r, (m + s m + s 1 Deotig the etire sum o the RHS of (22 by f 2 (r, it is straightforward to see that ( ( ( m + s m + r m + r + s f 2 (r + 1 f 2 (r (m + s m + s 1 It remais to verify the iitial coditio f 1 (0 f 2 (0; that is, (m 2 ( ( m+s m 2 ( m+2s ( m 1 m +s m + s ( + j (23 (m + s ( m+2s m+s ( + j + s m + s 1 Deote the summad o the LHS of (23 by F 2 (s, ad its sum by f 2 (s : F 2(s, Now, itroduce the compaio fuctio G 2 (s, : F 2 (s, 2 (m 2(m + s + 1

4 4 TEWODROS AMDEBERHAN, DAVID CALLAN, HIDEYUKI OHTSUKA AND ROBERTO TAURASO ad (routiely chec that F 2 (s + 1, F 2 (s, G 2 (s, + 1 G 2 (s, Telescopig gives f 2 (s + 1 f 2 (s F 2 (s + 1, F 2 (s, [G 2 (s, + 1 G 2 (s, ] ( ( m + s m + s G 2 (s, (m + s + 1 ( m Let F 3 (s, j be the summad o the RHS of (23 ad its sum f 3 (s : m 1 F 3 (s, j Itroduce j( m + j + 1 G 3 (s, j : F 3 (s, j ( + s + 1(m + s ad chec that F 3 (s + 1, j F 3 (s, j G 3 (s, j + 1 G 3 (s, j Summig ad telescopig, we get f 3 (s + 1 f 3 (s m 1 F 3 (s + 1, j m 1 F 3 (s, j m 1 ( ( m + s m + s G 3 (s, m 0 (m + s + 1 The iitial coditio f 2 (0 f 3 (0 is precisely the cotet of Lemma 21 [G 3 (s, j + 1 G 3 (s, j] ( m The ext statemet covers Problem [3] as a immediate applicatio of Theorem 22 Corollary 23 Let a, b ad c be o-egative itegers The, the fuctio ( a + b c 1 ( ( a + j b + j U(a, b, c : a a a b 1 is symmetric, ie U(σ(a, σ(b, σ(c U(a, b, c for ay σ i the symmetric group S 3 Proof If a, m 2a, r b a, s c a, the left-had side of Theorem 22 turs ito ( b+c a ( ( ( 2a,b a,c a LHS ( 2b ( 2c (2a 2 + b a + c a b+a c+a (a + b!(b + c!(c + a! a ( ( ( 2 (2a!(2b!(2c! a b c 2Q a,bq b,c Q c,a a ( ( ( Q a,a Q b,b Q c,c a + b + c + ad the right-had side simplifies to ( a + c b 1 ( ( a + j c + j b 1 RHS a aq c,a c a c 1 Therefore, we obtai (24 Q a,b Q b,c Q c,a Q a,a Q b,b Q c,c a ( ( ( aq c,a a + b + c + 2 ( a + j a b 1 ( a + j a ( c + j c 1 ( c + j c 1

5 The followig apparetly symmetry a ( ( ( a + b + c + mi{a,b,c} ( ( ( a + b + c + implies that the LHS of the idetity i (24 has to be symmetric The assertio follows from the symmetry iherited by the RHS of the same euatio (24 Example 24 I euatio (24, the special case a, b c m becomes (12 while a b, c m recovers (13 Corollary 25 Preserve otatios from Cor 23 For a, b, c N ad ay σ S 3, we have a ( ( ( a + b b + c c + a σ(aq σ(c 1 ( ( σ(a,σ(b σ(a + j σ(b + j (25 a + b + c + 2 σ(a σ(b 1 Proof First, employ a algebraic maipulatio o (24 similar to euatio (11 Now apply the idetity i (24 ad the statemet of Corollary 23 For o-egative itegers x, y, z, write the elemetary symmetric fuctios e 1 (x, y, z x + y + z, e 2 (x, y, z xy + yz + zx ad e 3 (x, y, z xyz Theorem 26 For o-egative itegers a, b ad c, we have a ( ( ( a + b b + c c + a 3 b2 c 2 Q b,c a 1 e 2 (a, b, c ( ( b+j c+j (26 b c a + b + c + 2 e 2 (j, b, c e 2 (j + 1, b, c Proof Oce agai use the W-Z method First, divide through by e 2 (a, b, c to deote the summad o the LHS of (26 by F 1 (a, ad its sum by f 1 (a : a F 1(a, Now, itroduce the compaio fuctio G 1 (a, : F 1 (a, ((e 2 + b + c 2 (e 2 + b + c + abc + bc(b + (c (a + 1 (e 2 + b + c ad (routiely chec that F 1 (a + 1, F 1 (a, G 1 (a, + 1 G 1 (a, ; where we write e 2 for e 2 (a, b, c Keepig i mid that F 1 (a, a ad telescopig gives a+1 a+1 a+1 f 1 (a + 1 f 1 (a F 1 (a + 1, F 1 (a, [G 1 (a, + 1 G 1 (a, ] 5 G 1 (a, a + 2 G 1 (a, 0 0 G 1 (a, 0 b2 c 2 Q a,b Q b,c Q c,a 2e 2 (e 2 + b + c This differece formula for f 1 (a + 1 f 1 (a leads to f 1 (a b2 c 2 Q b,c 2 which is the reuired coclusio a 1 ( b+j a ( c+j c (jb + bc + cj (jb + bc + cj + b + c

6 6 TEWODROS AMDEBERHAN, DAVID CALLAN, HIDEYUKI OHTSUKA AND ROBERTO TAURASO Remar 27 I [7], Miaa, Ohtsua ad Romero obtaied two idetities for the sum From Theorem 26 ad (11, we obtai the idetity for the sum a B a,b b, B c, B3, Remar 28 Corollary 25 ad Theorem 26 exhibit formulas for ( ad 3 ( It appears that similar (albeit complicated results are possible for sums of the type p ( wheever p is a odd positive iteger (but ot whe p is eve We ca offer a 4-parameter geeralizatio of Theorem 25 ad Theorem 26 Theorem 29 For o-egative itegers a, b, c ad d, we have a ( ( ( ( a + b b + c c + d d + a bq b,cq c,d Q b+c+d,a a + b + c + d + 2Q a,c Proof Aalogous to the precedig argumets a 1 Q b,j Q c 1,j+1 Q d 1,j+1 Q b+c+d,j+1 Remar 210 It is iterestig to compare our results agaist Corollary 41 of [5] Although these are similar, there are differeces: i our case the RHSs are less ivolved while those of [5] are more geeral See also Corollary 42 ad Theorem 43 of [7] The examples below are devoted to explore some specifics Example 211 Set a b c i Theorem 25 The outcome is ( 3 1 ( ( 2 + j 1 j Example 212 Set a b c i Theorem 26 The outcome is ( ( ( +j ( + 2j( + 2j + 2 Example 213 Set a b c d i Theorem 29 The outcome is ( 4 1 ( ( ( 3 ( j j j The secod set of mai results We start with a -idetity ad its ordiary couterpart will allow us to prove oe of the Mothly problems which was alluded to i the Itroductio Alog the way, we ecouter the Catala triagle t, ( ( 1 which we also write as t+, ( ( 1 Let s recall some otatios The -aalogue of the iteger is give by [] : 1, the 1 factorial by []! 1 i i1 ad the biomial coefficiets by 1 ( []! []![ ]! Lemma 31 For a free parameter ad a positive iteger, we have ( [ ( ( ] ( + 1 (+1 1 2

7 7 Proof Let G(, ( Now, chec that ( [ ( ( ] (+1 ( G(, G(, + 1 ad the sum over 0 through to obtai G(, 0 ( 2 We ow demostrate a combiatorial argumet for the special case 1 of Lemma 31 Lemma 32 For o-egative itegers, we have (31 ( [( ( ] ( 2 Proof The first factor i the summad o the left side of (31 couts paths of + 1 steps, cosistig of upsteps (1, 1 or dowsteps (1, 1, that start at the origi ad ed at height The secod factor is the geeralized Catala umber that couts oegative (ie, first uadrat paths of up/dow steps that ed at height 2 By cocateatig the first path ad the reverse of the secod, we see that the left side couts the set X of paths of + 1 upsteps ad dowsteps that avoid the x-axis for x >, ie avoid ( + 2, 0, ( + 4, 0,, (4, 0 Now ( is the umber of balaced paths of legth (ie, upsteps ad dowsteps, but it is also the umber of oegative -paths ad, for 1, twice the umber of positive ( oegative, o-retur -paths (see [6], for example So, the right side of (31 couts the set Y of pairs (P, Q of oegative -paths Here is a bijectio φ from X to Y A path P X eds at height 1 ad so its last upstep from the x-axis splits it ito P BUD where B is a balaced path ad D is a dyc path of legth sice P avoids the x-axis for x > Write D as QR where R is of legth If B is empty, set φ(p (Q, Reverse(R, a pair of oegative -paths edig at the same height If B is oempty, the by the above remars it is euivalet to a bicolored positive path S of the same legth, say colored red or blue If red, set φ(p (Q S, Reverse(R Y with the first path edig strictly higher tha the secod If blue, set φ(p (Reverse(R, Q S Y with the first path edig strictly lower tha the secod It is easy to chec that φ is a bijectio from X to Y As a applicatio, we preset a proof for Problem as advertised i the Itroductio Corollary 33 For o-egative positive iteger, we have ( ( ( ( ( ( 2

8 8 TEWODROS AMDEBERHAN, DAVID CALLAN, HIDEYUKI OHTSUKA AND ROBERTO TAURASO Proof Start by writig ( ( + 1 A 1 :, A 2 : +1 ( ( + 1 Ã 1 :, Ã 2 : ( ( 1 ( + 1 ( + 1 Re-idexig gives A 1 Ã1 ad A 2 Ã2 The reuired idetity is A 1 + Ã1 2A 1 ( 4+1 ( + 2 I view of the Vadermode-Chu idetity A1 + A 2 ( 4+1, it suffices to prove that A 1 A 2 A 1 Ã2 ( 2 That is, ( + 1 t, ( + 1 [( ( 1 ], ( 2 which is exactly what Lemma 32 is about However, here is yet aother verificatio: if we let G(, ( 2 + the it is routie to chec that ( [( ( ] ( G(, G(, Obviously the [G(, G(, + 1] G(, 0 G(, + 1 G(, 0 The proof follows 4 Cocludig Remars ( 2 Fially, we list biomial idetities with extra parameters similar to those from the precedig sectios, however their proofs are left to the iterested reader because we wish to limit uduly replicatio of our techiues We also iclude some ope problems The first result geeralizes Corollary 25 Propositio 41 For o-egative itegers a, b, c ad a iteger r, we have a+r ( ( ( c+r 1 a + b + r b + c + r c + a + r ( ( a + j b + j (2 r (a + rq a+r,b a + b + c + a b + r 1 1 Next, we state certai atural -aalogues of Corollary 25 ad Corollary 23 Theorem 42 For o-egative itegers a, b ad c, we have a ( ( ( ( ( a + b b + c c + a a + b 1 a a + b + c + a c 1 ( a + j j a Corollary 43 Let a, b ad c be o-egative itegers The, the fuctio U (a, b, c : 1 ( a a + b c 1 ( ( a + j b + j 1 a a b 1 ( b + j b 1

9 9 is symmetric, ie U (σ(a, σ(b, σ(c U (a, b, c for ay σ i the symmetric groups S 3 Let s cosider the family of sums S r (a, b, c : a ( ( ( a + b b + c c + a 2r+1 a + b + c + It follows that ( ( ( ( a + b c + a a + b c + a (a 2 2 (a + (a a + c + a + a ( ( a 1 + b c + a 1 (a + b(a + c a 1 + a 1 ( ( a 1 + b c + a 1 (a + b(a + c a 1 + c + which i tur implies, after replacig 2r+1 2r 1 2 2r 1 [a 2 (a 2 2 ], that S r (a, b, c a 2 S r 1 (a, b, c (a + b(a + c S r 1 (a 1, b, c Problem Itroduce the operators o symmetric fuctios f f(a, b, c of 3-variables by L f [(a + b(a + ce a 2 I]f where E f(a, b, c f(a 1, b, c ad I f(a, b, c f(a, b, c is the idetity map As a uestio of idepedet iterest show that the iterates L 1 always yield i symmetric polyomials i Z[a, b, c], for ay iteger 1 Postscript Matthew Hogye Xie of Naai Uiversity iformed the authors, i private commuicatio, that he has foud a proof for this problem Cojecture 44 For each r Z +, there exist symmetric polyomials f r, g r Z[a, b, c] such that S r (a, b, c b2 c 2 f r (a, b, c ( b+c a 1 ( b+j ( c+j b b c gr (j + 1, b, c 2 f r (j, b, c f r (j + 1, b, c The fuctios f r satisfy the recurrece, with f 0 (a, b, c 1 f r (a, b, c L f r 1 (a, b, c r f r g r 0 1 1/e 3 1 e e 2 2 e 1 e 2 + e 3 2e 3 e 2 3 e e 3 e 2 3e 2 2e 1 2e 3 e 1 + 2e 2 e e 3 e 2 e 1 6e 2 3 8e 3 e 2 + 3e e 3 e 2 e 1 Table 1 The first few polyomials i support of Cojecture 44

10 10 TEWODROS AMDEBERHAN, DAVID CALLAN, HIDEYUKI OHTSUKA AND ROBERTO TAURASO Refereces [1] P 11844, The Amer Math Mothly, 122, (May 2015, [2] P 11899, The Amer Math Mothly, 123, (March 2016, [3] P 11916, The Amer Math Mothly, 123, (Jue-July 2016, [4] R P Bret, H Ohtsua, J-A H Osbor, H Prodiger, Some biomial sums ivolvig absolute values, J Iteger Se, 19 (2016, A1637 [5] V J W Guo, J Zeg, Factors of biomial sums from the Catala triagle, J Numb Theory, 130 (2010, [6] David Calla, Bijectios for the idetity 4 ( 2 ( 2(, upublished, available at [7] P J Miaa, H Ohtsua, N Romero, Sums of powers of Catala triagle umbers, preprit available at [8] M Petov se, H Wif, D Zeilberger, AB, A K Peters/CRC Press, 1996 Departmet of Mathematics, Tulae Uiversity, New Orleas, LA 70118, USA address: tamdeber@tulaeedu Departmet of Statistics, Uiversity of Wiscosi-Madiso, Madiso, WI 53706, USA address: calla@statwiscedu Buyo Uiversity High School, , Kami, Ageo-city, Saitama Pref, , Japa address: otsuahideyui@gmailcom Dipartimeto di Matematica, Uiversita di Roma Tor Vergata, Via della Ricerca Scietifica, 1, Roma, Italy address: tauraso@matuiroma2it