ACTA ARITHMETICA 1271 (2007 Factors of alteratig sums of products of biomial ad q-biomial coefficiets by Victor J W Guo (Shaghai Frédéric Jouhet (Lyo ad Jiag Zeg (Lyo 1 Itroductio I 1998 Cali [4 proved that for all positive itegers m ad ( 2 1 ( 2 m (11 ( 1 + is a iteger by arithmetical techiques For m 1 2 ad 3 by the biomial theorem Kummer s formula ad Dixo s formula it is easy to see that (11 is equal to 0 1 ad ( 3 respectively Recetly i the study of fiite forms of the Rogers Ramauja idetities [7 we stumbled across (11 for m 4 ad m 5 which gives ( ( 2 + 2 + 0 2 ad j 0 ( 3 ( 2 + ( 2 2 + respectively Ideed de Bruij [3 has show that for m 4 there is o closed form for (11 by asymptotic techiques Our first objective is to give a q-aalogue of Cali s result which also implies that (11 is positive for m 2 I 2004 Zudili [14 proved that for all positive itegers j ad r ( 2 1 ( ( ( 2j ( 1 2 + 1 2 + j r (1 Z j + + 1 j which was origially observed by Strehl [12 i 1994 I fact Zudili s motivatio was to solve the followig problem which was raised by Schmidt [11 i 1993 ad was apparetly ot related to Cali s result 2000 Mathematics Subject Classificatio: 05A10 05A30 11B65 [17
18 V J W Guo et al Problem 11 (Schmidt [11 For ay iteger r 2 defie a sequece of umbers {c (r } N idepedet of the parameter by ( r ( + r ( ( + c (r 0 Is it true that all the umbers c (r are itegers? At the ed of his paper Zudili [14 raised the problem of fidig ad solvig a q-aalogue of Problem 11 Our secod objective is to provide such a q-aalogue For ay iteger defie the q-shifted factorial (a by (a 0 1 ad { (1 a(1 aq (1 aq 1 1 2 (a ((1 aq 1 (1 aq 2 (1 aq 1 1 2 0 We will also use the compact otatios for m 1: (a 1 a m : (a 1 (a m (a 1 a m : lim (a 1 a m The q-biomial coefficiets are defied as [ : [ q (q (q (q Sice 1/(q 0 if < 0 we have [ 0 if > or < 0 The followig is our first geeralizatio of Cali s result Theorem 12 For m 3 ad all positive itegers 1 m 1 (13 ( 1 q (m 12+( m i + i+1 1 i1 i + 1 + m m 2 λi 1 i+1 + q λ2 i i+2 i+1 λ i 1 where m+1 λ 0 1 ad the sum is over all sequeces λ (λ 1 λ m 2 of oegative itegers such that λ 0 λ 1 λ m 2 Cali [4 has give a partial q-aalogue of (11 by cosiderig the alteratig sum 0 ( 1 q j[ m I this respect besides (13 we shall also prove the followig divisibility result Theorem 13 For all positive itegers 1 m m+1 1 the alteratig sum 1 + 1 m 1 S( 1 m ; j q : ( 1 q j2+( m i + i+1 1 1 i1 i + is a polyomial i q with oegative itegral coefficiets for 0 j m 1 λ i1 λ i
Biomial ad q-biomial coefficiets 19 We shall give two proofs of Theorem 12 The first oe is based o a recurrece relatio formula for S( 1 m ; j q which also leads to a proof of Theorem 13 The secod oe follows directly from Adrews basic hypergeometric idetity betwee a sigle sum ad a multiple sum [1 Theorem 4: Theorem 14 (Adrews [1 For every iteger m 0 the followig idetity holds: (a q a q a b 1 c 1 b m c m q N (14 (q a a aq/b 1 aq/c 1 aq/b m aq/c m aq N+1 0 ( a m q m+n b 1 c 1 b m c m (aq aq/b mc m N (aq/b 1 c 1 l1 (aq/b m 1 c m 1 lm 1 (aq/b m aq/c m N (q l1 (q lm 1 l 1 l m 1 0 (b 2 c 2 l1 (b m c m l1 + +l m 1 (aq/b 1 aq/c 1 l1 (aq/b m 1 aq/c m 1 l1 + +l m 1 (q N l1 + +l m 1 (b m c m q N /a l1 + +l m 1 (aq lm 2+ +(m l1 q l1+ +lm 1 (b 2 c 2 l 1 (bm 1 c m 1 l 1+ +l m 2 Note that there are two igrediets i Zudili s approach to Problem 11: oe is Theorem 14 with q 1 ad the other is the Legedre trasform Now a q-legedre trasform reads: (15 a 0 q ( 2 + b b 1 q2+1 ( 1 1 q ++1 0 2 a which is a special case of Carlitz s q-gould Hsu iverse formula [5 (see also [10 Usig (15 ad Theorem 14 i its full geerality we are able to formulate ad prove a q-aalogue of Problem 11 Theorem 15 For ay iteger r 1 defie ratioal fractios c (r (q of the variable q idepedet of by writig (16 q r( 2 +(1 r( r + r 0 q ( 2 +(1 r( + c (r (q The c (r (q N[q 0 Remar Sice the r 1 case is trivial we suppose that r 2 i what follows
20 V J W Guo et al As [ that [ + [ 2 [ + ivoig (15 we derive immediately from (16 2 2j r q (1 r( c (r (q t (r j j (q j j0 where t (r j (q qr(j+1 2 [ 1 q2+1 2 ( 1 1 q ++1 [ + j j r q ( rj Therefore Theorem 15 is a cosequece of the followig theorem which is our q-aalogue of Zudili s result (1 Theorem 16 For ay iteger r 2 2j 2 1 q (r 1( t (r j (q N[q j As will be show Theorem 16 follows directly from Adrews idetity (14 This paper is orgaized as follows We prove Theorems 12 ad 13 i the ext sectio The proof of Theorem 16 is give i Sectio 3 Some iterestig divisibility results are give i Sectio 4 I the last sectio we preset four related cojectures 2 Proof of Theorems 12 ad 13 We will eed two ow idetities i q-series Oe is the q-pfaff Saalschütz idetity [6 Appedix (II1 (see also [8 13: 1 + 2 2 + 3 3 + 1 (21 1 + 2 + 3 + 1 r0 q 2 +2r (q 1 + 2 + 3 r (q r (q r+2 (q 1 r(q 2 r(q 3 r where 1/(q 0 if < 0 ad the other is the q-dixo idetity: 1 (2 ( 1 q (32 /2 1 + 2 2 + 3 3 + 1 1 + 2 + 3 + 1 A short proof of (2 is give i [9 We first establish the followig recurrece formula (q 1 + 2 + 3 (q 1 (q 2 (q 3
Biomial ad q-biomial coefficiets 21 Lemma 21 Let m 3 The for all positive itegers 1 m ad ay iteger j the followig recurrece holds: 1 1 2 + (23 S( 1 m ; j q q l2 3 S(l 3 m ; j 1 q l 2 l l0 Proof For ay iteger ad positive itegers a 1 a l let l ai + a i+1 C(a 1 a l ; a i + where a l+1 a 1 The (24 S( 1 m ; j q (q 1 (q m ( 1 q j2+( C(1 m ; (q 1 + m 1 We observe that for m 3 we have C( 1 m ; (q 2 + 3 (q m + 1 (q 1 + 2 (q m + 3 ad by lettig 3 i (21 [ 1 + 2 1 + [ 1 + 2 2 + 1 r0 i1 1 [ 1 + 2 1 + 1 + 2 C( 3 m ; 2 + q r2 +2r (q 1 + 2 (q r (q r+2 (q 1 r(q 2 r Pluggig these ito (24 we ca write its right-had side as 1 R: 1 1 r0 ( 1 C( 3 m ; q(r+2 +(j 1 2 +( (q2 + 3 (q 1 (q m (q r (q r+2 (q 1 r(q 2 r(q m + 3 Settig l r + we see that 1 l 1 but if l < 0 at least oe of the idices l + ad l is egative for ay iteger which implies that 1/(q l (q l+ 0 by covetio Therefore exchagig the order of summatio we have R 1 l0 q l2 (q 2 + 3 (q 1 (q m (q 1 l(q 2 l(q m + 3 l +( ( 1 C( 3 m ; q(j 12 (q l (q l+ l Now i the last sum maig the substitutio C( 3 m ; (q l (q l+ (q m + 3 (q 3 +l(q m +l we obtai the right-had side of (23 (25 C(l 3 m ; First proof of Theorem 12 Lettig 3 i (2 yields S( 1 2 ; 1 q 1 Theorem 12 the follows by iteratig m 2 times formula (23
22 V J W Guo et al Secod proof of Theorem 12 Sice M M (q ( 1 q (M N ( M+N N + N (q N+1 by collectig the terms of idex ad the left-had side of (13 ca be writte as m i + i+1 1 L : + (1 + q ( 1 q (m 12 +( m i + i+1 i i + i1 1 i1 m { i + i+1 1 1+ (1+q ( 1 (m 1 q (m 1(+1 2 m } q i (q i+1 i (q i+1 i1 1 Lettig c 1 c m c ad a 1 i Adrews formula (14 we get (26 1 + + q 1(1 (b 1 b m q N ( q (q/b 1 q/b m q N+1 ( 1 m m+n q m( b 1 b m (q N (q/b m N i1 l 1 l m 1 0 m 1 ( (b i+1 l1 + +l i 1 (q/b i l1 + +l i Now shiftig m to m 1 i (26 settig i1 (q N i1 (q l1 (q lm 1 (q N l1 l m 1 b i+1 l1 + +l i q (l 1 + +l i 2 +(m il i N m b i q i for i 1 m 1 ad λ i l 1 + + l i for i 1 m 2 oe sees that L equals m i + i+1 (q 2 m 2 m (q i+1 λi ( 1 λ i q (λ i +1 2 + i+1 λ i i (q 1+ m 1 m (q 1+ i λi (q λi λ i 1 0 λ 1 λ m 2 i1 1 + m 1 0 λ 1 λ m 2 m 2 i1 q λ2 i [ λi+1 λ i [ i + i+1 i + λ i where λ 0 0 ad λ m 1 m The last idetity is clearly equivalet to that i Theorem 12 I order to prove Theorem 13 we shall eed the followig relatio: (27 S( 1 m ; 0 q S( 1 m ; m 1 q 1 q 1 2 + 2 3 + + m 1 m As [ q [ q ( equatio (27 ca be verified by substitutig q by 1 q 1 ad the replacig by i the defiitio of S( 1 m ; m 1 q
Biomial ad q-biomial coefficiets 23 Proof of Theorem 13 We proceed by iductio o m 1 By the q- biomial theorem [6 Appedix (II3 we have 1 2 S( 1 ; 0 q ( 1 q ( 0 + 1 I view of (25 it follows from (27 that S( 1 2 ; 0 q S( 1 2 ; 1 q 1 q 1 2 q 1 2 So the theorem is valid for m 2 Now suppose that the expressio S( 1 m 1 ; j q is a polyomial i q with oegative itegral coefficiets for some m 3 ad 0 j m 2 The by the recurrece formula (23 so is S( 1 m ; j q for 1 j m 1 It remais to show that S( 1 m ; 0 q has the required property Sice the q-biomial coefficiet [ is a polyomial i q of degree ( (see [2 p 33 it is easy to see from the defiitio that the degree of the polyomial S( 1 m ; m 1 q is less tha or equal to 1 2 + 2 3 + + m 1 m It follows from (27 that S( 1 m ; 0 q is also a polyomial i q with oegative itegral coefficiets This completes the iductive step of the proof Remar Though it is ot ecessary to chec the m 3 case to validate our iductio argumet we thi it is coveiet to iclude here the formulas for m 3 First the q-dixo idetity (2 implies that 1 + 2 + 3 S( 1 2 3 ; 1 q From (23 ad (25 we derive 1 S( 1 2 3 ; 2 q Fially applyig (27 we get l0 q l2 [ 1 l 2 2 + 3 2 l S( 1 2 3 ; 0 q S( 1 2 3 ; 2 q 1 q 1 2 + 2 3 1 l0 [ q ( 1 l( 2 l+ 3 l 1 l [ 2 + 3 2 l 3 Proof of Theorem 16 We will distiguish the cases where r 2 is eve or odd ad treat separately the values r 2 ad r 3
24 V J W Guo et al For r 2 apply (14 specialized with m 1 a q (2+1 N j b 1 q ad c 1 q ( j The left-had side of (14 is the equal to [ + j 2j 2 q 2( j 2 +( t ( j (q Equatig this with the right-had side gives t ( j (q which shows that [ 2j j (q 2 (q 2 j (q (q 2j (q 2j (q 2 j [ 2 1q ( ( t (q N[q j q 2( j 2 ( For r 3 apply (14 specialized with m 1 a q (2+1 N j ad b 1 c 1 q ( j This yields i that case which shows that [ 2j j t (3 j (q (q 2 (q 3j (q 3 j [ 2 1q 2( (3 t (q N[q j q 3( j 2 2( For r 2s 4 apply (14 with m s 2 a q (2+1 N j b 1 q ad c 1 b i c i q ( j for all i {2 s} to get q (2s 1( 2s( j 2 t (2s j (q (q 2 (q j (q (q 2j (q j l 2 0 [ l s 1 0 [ 2j l 2 l 1 0 [ j l1 l 1 j l1 + j q (l 1 2 +2j(s 1l 1 +(j+1 l 1 l 1 j l1 l 2 + j 2 q (l 2 +2j(s l 2 +(j+1 l 2 l 1 l 2 j [ 2j l s 1 l1 l s 1 + j 2 q (l s 1 2 +2jl s 1+(j+1 l s 1 l 1 l s 1 j 2j l 1 l s 1 j q (l 1 + +l s 1 2 As the coditio l 1 + + l s 1 j holds i the last summatio we ca see that for s 2 [ 2j 2 1q (2s 1( (2s j t j (q N[q For r 2s+1 5 apply (14 with m s 2 a q (2+1 N j ad b i c i q ( j for all i {1 s} to get
q 2s( (2s+1( j Biomial ad q-biomial coefficiets 25 2 (2s+1 t j (q (q 2 2j l1 + j 2 (q 2j (q 2 q (l 1 +2j(s 1l 1 +(j+1 l 1 j l 1 l 1 j l 2 0 [ l s 1 0 [ 2j l 2 l 1 0 l1 l 2 + j 2 q (l 2 +2j(s l 2 +(j+1 l 2 l 1 l 2 j [ 2j l s 1 l1 l s 1 + j 2 q (l s 1 2 +2jl s 1+(j+1 l s 1 l 1 l s 1 j 2j l 1 l s 1 j j0 q (l 1 + +l s 1 2 As the coditio l 1 + + l s 1 j holds i the last summatio we ca see that for s 2 [ 2j 2 1q 2s( (2s+1 j t j (q N[q Remar I the special case r 2 our proof gives the followig expressio for the coefficiets c ( (q: 2j 2 (31 c ( (q q 2( j 2 j These coefficiets are q-aalogues of the famous c ( (1 ivolved i Apéry s proof of the irratioality of ζ(3: (3 c ( (1 j0 ( 2j ( 2 j j0 ( 3 j As explaied i [12 whe q 1 oe ca derive the last expressio from (31 i a elemetary way (by two iteratios of the Chu Vadermode formula But our q-aalogue (31 does ot lead to a atural q-aalogue of (3 4 Cosequeces of Theorems 12 ad 13 Lettig q 1 i Theorem 12 we obtai a direct geeralizatio of Cali s result (11 Theorem 41 For m 3 ad all positive itegers 1 m 1 m ( (41 ( 1 i + i+1 i + 1 i1 ( 1 + m m 2 ( ( λi 1 i+1 + i+2 1 λ i i+1 λ i λ i1
26 V J W Guo et al where m+1 λ 0 1 ad the sum is over all sequeces λ (λ 1 λ m 2 of oegative itegers such that λ 0 λ 1 λ m 2 Remar For m 1 ad 2 it is easy to see that the left-had side of (41 is equal to 0 ad ( 1 + 2 1 respectively Cali s result follows from (41 by settig i for all i 1 m Lettig 1 m i Theorem 13 we obtai a complete q-aalogue of Cali s result Corollary 42 For all positive m ad 0 j m 1 2 1 ( 1 q j2 +( 2 m + is a polyomial i q with oegative itegral coefficiets Lettig 2i 1 m ad 2i for 1 i r i Theorem 13 we obtai Corollary 43 For all positive m r ad 0 j 2r 1 m + 1 m ( 1 q j2 +( m + r m + r m m + + m is a polyomial i q with oegative itegral coefficiets I particular m ( m + r ( m + r ( 1 m + + m is divisible by ( m+ m Lettig 3i 2 l 3i 1 m ad 3i for 1 i r i Theorem 13 we obtai Corollary 44 For all positive l m r ad 0 j 3r 1 l + 1 l ( 1 q j2 +( l + m r m + r + l r l + m + + l is a polyomial i q with oegative itegral coefficiets I particular l ( l + m r ( m + r ( + l r ( 1 l + m + + is divisible by ( l+m l l ( m+ m ad ( +l Lettig m 2r + s 1 3 2r 1 + 1 ad all the other i be i Theorem 41 we get
Biomial ad q-biomial coefficiets 27 Corollary 45 For all positive r s ad ( 2 + 1 r ( 2 + 1 r ( 2 ( 1 + + 1 + + is divisible by both ( ( ad 2+1 ( ad is therefore divisible by (2+1 2 However the followig result is ot a special case of Theorem 41 Corollary 46 For all oegative r ad s ad positive t ad ( 2 + 1 r ( 2 + 1 s ( 2 t ( 1 + + 1 + + is divisible by ( 2 Proof We proceed by iductio o r s The r s case is clear from Corollary 45 Suppose the statemet is true for r s m 1 By Theorem 41 oe sees that ( 2 + 2 m ( 2 + 1 s ( 2 + 1 s ( 2 t (4 ( 1 + + 1 + + 1 + + 2 + 2 ( 2 + 2 m 1 ( 2 + 1 s+1 ( 2 + 1 s+1 ( 2 t 1 ( 1 2 + 1 ++1 ++1 + + where m t 1 is divisible by ( ( 2 + 2 2 + 1 2 2 2 + 1 By the biomial theorem we have ( 2 + 2 m (( ( 2 + 1 2 + 1 m (43 + + + 1 + + 1 + m ( ( m 2 + 1 i ( 2 + 1 m i i + + 1 + i0 Substitutig (43 ito the left-had side of (4 ad usig the iductio hypothesis ad symmetry we fid that ( 2 + 1 m+s ( 2 + 1 s ( 2 t ( 1 + + 1 + + ( 2 + 1 s ( 2 + 1 m+s ( 2 t + ( 1 + + 1 + + s
28 V J W Guo et al is divisible by 2 ( 2 However replacig by oe sees that ( 2 + 1 m+s ( 2 + 1 s ( 2 t ( 1 + + 1 + + ( 2 + 1 s ( 2 + 1 m+s ( 2 t ( 1 + + 1 + + This proves that the statemet is true for r s m It is clear that Theorems 13 ad 41 ca be restated i the followig forms Theorem 47 For all positive itegers 1 m ad 0 j m 1 the alteratig sum m (q i + (q i+1 1 1 ( 1 q j2+( m 2i (q i1 2i 1 i1 i + where m+1 0 is a polyomial i q with oegative itegral coefficiets Theorem 48 For all positive itegers 1 m we have m ( i + i+1! 1 m ( 1! ( 1 2i N (2 i! i + i1 1 i1 where m+1 0 It is easy to see that for all positive itegers m ad the expressio (2m!(!/(m +!m!! is a iteger by cosiderig the power of a prime dividig a factorial Lettig 1 r m ad r+1 r+s i Theorem 48 we obtai Corollary 49 For all positive m r ad s m ( 2m r ( 2 ( 1 m + + m is divisible by (2m!(!/(m +!m!! I particular we fid that ( 4 r ( 2 ( 1 2 + + is divisible by ( 4 ad is divisible by (6!(!/(4!(3!! ( 6 r ( 2 ( 1 3 + + s s s
Biomial ad q-biomial coefficiets 29 From Theorem 48 it is easy to see that m ( i + i+1! 1 m ( ri 1! ( 1 2i (2 i! i + i1 1 i1 where m+1 0 is a oegative iteger for all r 1 r m 1 For m 3 lettig ( 1 2 3 be ( 3 (2 3 or (2 4 we obtai the followig two corollaries Corollary 410 For all positive r s t ad ( 6 r ( 4 s ( 2 ( 1 3 + 2 + + is divisible by both ( ( 6 ad 6 3 Corollary 411 For all positive r s t ad ( 8 r ( 4 s ( 2 ( 1 4 + 2 + + is divisible by ( 8 3 5 Some ope problems Based o computer experimets we would lie to preset four iterestig cojectures The first two are refiemets of Corollaries 410 ad 411 respectively Cojecture 51 For all positive r s t ad ( 6 r ( 4 s ( 2 t ( 1 3 + 2 + + is divisible by both 2 ( 6 ( ad 6 6 3 Cojecture 52 For all positive r s t ad with (r s t (1 1 1 ( 8 r ( 4 s ( 2 t ( 1 4 + 2 + + is divisible by 2 ( 8 3 Cojectures 51 ad 52 are true for r + s + t 10 ad 100 Let gcd(a 1 a 2 deote the greatest commo divisor of itegers a 1 a 2 Cojecture 53 For all positive m ad we have ( ( 2 r (51 gcd ( 1 r m m + 1 + t t ( 2
30 V J W Guo et al Let d 2 be a fixed iteger Every oegative iteger ca be uiquely writte as i 0 a i d i where 0 a i d 1 for all i ad oly fiitely may umbers of b i are ozero deoted by [ a 1 a 0 d i which the first 0 s are omitted Let [a 1 a r 3 [b 1 b s 7 [c 1 c t 13 We ow defie three statistics α( β( ad γ( as follows: Let α( be the umber of discoected 2 s i the sequece a 1 a r Here two ozero digits a i ad a j are said to be discoected if there is at least oe 0 betwee a i ad a j Let β( be the umber of 1 s i b 1 b s which are ot immediately followed by a 4 5 or 6 Let γ( be the umber of 1 s i c 1 c t which are immediately followed by oe of 7 12 or immediately followed by a umber of 6 s ad the by oe of 7 12 For istace [20212 3 185 ad so α(185 2; [10142 7 2480 ad so β(2480 1; [1667 13 3296 ad so γ(3296 1 The first such that α( 4 is [2020202 3 1640; the first such that β( 4 is [1111 7 400; while the first such that γ( 4 is [17171717 13 97110800 We ed this paper with the followig cojecture Cojecture 54 For every positive iteger ( ( 2 3r gcd ( 1 r 1 2 + ( ( 2 3r+1 gcd ( 1 r 1 2 + ( ( 2 3r+2 gcd ( 1 r 1 2 + ( 2 3 α( ( 2 7 β( 13 γ( ( 2 Acowledgmets This wor was partially doe durig the first author s visit to Istitut Camille Jorda Uiversité Claude Berard (Lyo 1 ad was supported by a Frech postdoctoral fellowship We tha Christia Krattethaler for coveyig us his feelig that Theorem 12 should follow from Adrews formula [1 Theorem 4 We also tha Wadim Zudili for useful coversatios durig his visit i Lyo Fially the first author is grateful to Hog-Xig Dig for helpful commets o Cojecture 52
Biomial ad q-biomial coefficiets 31 Refereces [1 G E Adrews Problems ad prospects for basic hypergeometric fuctios i: Theory ad Applicatio of Special Fuctios R A Asey (ed Math Res Ceter Uiv of Wiscosi Publ No 35 Academic Press New Yor 1975 191 224 [2 The Theory of Partitios Cambridge Uiv Press Cambridge 1998 [3 N G de Bruij Asymptotic Methods i Aalysis Dover New Yor 1981 [4 N J Cali Factors of sums of powers of biomial coefficiets Acta Arith 86 (1998 17 26 [5 L Carlitz Some iverse relatios Due Math J 40 (1973 893 901 [6 G Gasper ad M Rahma Basic Hypergeometric Series 2d ed Ecyclopedia Math Appl 96 Cambridge Uiv Press Cambridge 2004 [7 V J W Guo F Jouhet ad J Zeg New fiite Rogers Ramauja idetities preprit 2006 arxiv: mathco/0604407 [8 V J W Guo ad J Zeg A combiatorial proof of a symmetric q-pfaff Saalschütz idetity Electro J Combi 12 (2005 #N2 [9 A short proof of the q-dixo idetity Discrete Math 296 (2005 259 261 [10 C Krattethaler A ew matrix iverse Proc Amer Math Soc 124 (1996 47 59 [11 A L Schmidt Geeralized q-legedre polyomials J Comput Appl Math 49 (1993 243 249 [12 V Strehl Biomial idetities combiatorial ad algorithmical aspects Discrete Math 136 (1994 309 346 [13 D Zeilberger A q-foata proof of the q-saalschütz idetity Europea J Combi 8 (1987 461 463 [14 W Zudili O a combiatorial problem of Asmus Schmidt Electro J Combi 11 (2004 #R22 Departmet of Mathematics East Chia Normal Uiversity Shaghai 200062 People s Republic of Chia E-mail: jwguo@mathecueduc http://mathecueduc/ jwguo Istitut Camille Jorda Uiversité Lyo 1 Bâtimet du Doye Jea Bracoier 43 blvd du 11 ovembre 1918 F-69622 Villeurbae Cedex Frace E-mail: jouhet@mathuiv-lyo1fr http://mathuiv-lyo1fr/ jouhet zeg@mathuiv-lyo1fr http://mathuiv-lyo1fr/ zeg Received o 22122005 (5110