INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 2008, #A05 Q-BINOMIALS AND THE GREATEST COMMON DIVISOR Keith R. Slavi 8474 SW Chevy Place, Beaverto, Orego 97008, USA slavi@dsl-oly.et Received: 7/21/07, Revised: 1/4/08, Accepted: 1/8/08, Published: 1/29/08 Abstract A q-biomial theorem is proved for q at complex roots of uity. This theorem comprises the Greatest Commo Divisor GCD fuctio. This theorem is the used to derive other ew product theorems, ad to express the GCD fuctio ad the GCD-sum fuctio as fiite products. 1. Itroductio The well-ow q-biomial or Gaussia coefficiet for a iteger 0 ca be defied [1, Sectio 3.3] as [ ] 1 q + 1 1 q 1 q 1 for 0, ad is defied to be zero outside this rage. It is show here i the Appedix that whe q e 2iπm/ q at roots of uity, we ca obtai a q-biomial ratioal root theorem: Theorem q-biomial ratioal root [ ], m if m, m/ e 2iπm/ 0 otherwise, m, Z, > 0, 0. 2 a where, m is the GCD of, m, ad is the ormal biomial fuctio. b
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 2008, #A05 2 The appearace of the GCD fuctio is surprisig, ad has may ramificatios i later results. 2. Fiite Product Theorems usig the GCD The well ow q-biomial aalog of the Newto biomial formula [1, Eq. 3.3.6] is give by 1 [ ] 1 + q t q 1/2 t. For q e 2iπm/, this becomes 1 1 + e 2iπm/ t e iπm 1 [ q ] e 2iπm/ t. 3 From the q-biomial ratioal root theorem 2, the q-biomial terms i the above summatio are o-zero whe m, ad the other terms are zero. If m ad trivially, the, m. Therefore a iterator r related to iterator is always iteger if r, m m. 4 We ca ow chage the iterator to r i the right-had side of 3, obtaiig 1 1 + e 2iπm/ t,m r0 [ e iπmr,m r,m 1 r/, m ] e 2iπm/ t r/,m. 5 We have already costraied m, so that from 4, rm/m, which is trivially true as m, m. Therefore the q-biomial ratioal root theorem 2 gives [ ], m. 6 r/, m r e 2iπm/ mr We substitute 6 ito right-had side of 5 ad, as r 1 is a iteger, we also,m,m substitute e iπ 1, obtaiig the followig biomial GCD theorem: 1,m 1 + e 2iπm/ t 1 mr r0,m r,m 1, m r t r/,m. 7 The above result ca give a very efficiet expasio whe, m. I additio, if is odd, the /, m is odd, ad therefore rr/, m 1 is always eve, so we get the special case 1 1 + e 2iπm/ t,m r0, m r t r/,m 1 + t /,m,m odd 1 8
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 2008, #A05 3 where the last step is from the biomial theorem. If we set t 1 we get 1 1 + e 2iπm/ 2,m odd 1. Iterestigly, this result gives a equatio for the GCD for odd as 1, m log 2 1 + e 2iπm/ odd 1. 9 Note that the right-had side ca be evaluated for m i the complex domai, although its iterpretatio for o-iteger values is uclear. From 9, usig cosx e ix + e ix /2, we get the real product, m + log 2 1/2 1 The GCD-sum g [2] ca be foud from 9 as cos mπ 2 odd 1. 1 1 1 g, m log 2 1 + e 2iπm/ odd 1. m0 m0 Note that g 2 1 is oly true if is prime, although the two-dimesioal product comprises a very iefficiet primality test. Substitutig t e 2ix ito 7 we ca obtai the followig trigoometric biomial theorem: 1 cos x + mπ 1 1 m, m 2 4 if, m is eve, m/2 + 0 otherwise 2,m 2 1 1 mr 2 1,m 2,m r, m cos 1 2r 1mπ x +, r, m 2 r0 10 for 2 ad eve. Similarly, substitutig t e 2ix ito 8, we ca also obtai: 1 cos x + mπ 1 1,m 2, m cos 1 2r 1mπ x +, 2 1 r, m 2 r0 11 for 1 ad odd. Similar results for sie products are obtaied by substitutig x x π/2.
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 2008, #A05 4 3. Proof of Some Well-ow Special Cases Proof. If, m 1, the both 10 ad 11 simplify to give the same special case for > 0 as 1 cos x + mπ 1 1mπ cos x + > 0,, m 1. 12 2 1 2 By settig m 1 ad substitutig x x π/2, equatio 12 gives 1 si x 2 1 si x + π. By settig m 1 ad substitutig x x π/2 π/2, equatio 12 gives 1 cos x 2 1 2 + 1π si x +. 2 Both results are also give i [3, Sec. 1.392]. If, m 2, the is eve, ad 10 applies to give the special case 1 cos x + mπ 1 1 m 1mπ 2 1 4 + cos x + > 0,, m 2. 2 Subcase m 2 of this special case is give i [3, Sec. 1.393], where the result has simply bee characterized as true for eve. Noe of these published special cases have eve hited at the more geeral coectio with the GCD fuctio. Acowledgmets Some of this wor was doe while I was worig for Tetroix, Ic. Their support is gratefully acowledged. Also I would lie to tha Professor George E. Adrews for his isight ad helpful commets. Refereces [1] George E. Adrews, The Theory of Partitios. Cambridge Uiversity Press, 1984, Chapter 3. [2] Kevi A. Brougha, The gcd-sum fuctio. Joural of Iteger Sequeces, Vol. 4 2001, Article 01.2.2. [3] I.S.Gradshtey ad I.M.Ryzhi, Table of Itegrals, Series, ad Products. 6th editio Academic Press, 2000.
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 2008, #A05 5 Appedix: Proof of Q-Biomial GCD Coefficiet Theorem We prove the q-biomial root of uity theorem 2, restated below [ ], m if m, m/, m, Z, > 0, 0. e 2iπm/ 0 otherwise 13 Proof. We start by usig 1 with q e 2imπ/, ad evaluatig it as a limit as x mπ/ : [ ] 1 e 2i +x lim T x πm/ 1 e 2ix 14 where the T are defied by e 2iπm/ T lim x mπ/ 1 1 1 e 2i +x. 15 1 e 2ix We ow prove 13 for each of the two cases m ad m separately. Case 1 : m The proof of this case is split ito two further subcases: the product terms T i 14 where m, ad the other terms where m. The two sets of products are later recombied to obtai the product for the case m. Subcase 1.1: m, m We first cosider the value of T for all m i the product 14 where m. I this case, at the limit, the deomiator of T i 15 is ever zero, so we evaluate at the limit ad separate out the term i the umerator to get T e 2iπ m/ e 2iπm/ 1 e 2iπm/ m, m. 16 Now m so therefore m/ is a iteger ad therefore e 2iπ m/ 1. As m, oe of the terms are zero ad 16 simplifies to Subcase 1.2: m, m T m, m. 17 We ext cosider the value of T for all i the product 13 where m ad m. I this case, m/ ad + m/ are both itegers so the umerator ad deomiator of 15 both ted to zero at the limit. We use L Hospital s rule, taig the ratio of the partial derivatives of the umerator ad deomiator with respect to x to give T 2i + e 2i +x lim x πm/ 2ie 2ix + m, m. 18
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 2008, #A05 6 Now we combie subcases 1.1 ad 1.2 to obtai T whe m. We first split the product of 14 ito two separate products, oe where m, ad the other where m. Usig results 17 ad 18, we obtai a sigle product over the rage 1 as [ ] e 2iπm/ m 1 +. 19 Comparig 19 with 13, it remais to prove oly that, m, m/ m + m, 0. 20 Note that this product oly cotais those terms for which m. As 0, we ca apply a chage of variable + to the right-had side of 20: + m + m m + 1 + m m For the case m, the above coditio + m is equivalet to m. Therefore. m + + 1 m m m m m f f f. 21 where the fuctio fp p m. We ow have eed of the followig. Lemma 1 Let r be a positive iteger such that r, m. 22
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 2008, #A05 7 If p is a iteger such that the r p ad p m pm 23 p/r qr. 24 q1 Proof. We first assume that qr, ad the show that the set of terms i each product are the same. From 22 we get q/, m. 25 For ay iteger iterator value q i the rage give i the right-had product, we eed to show that m is true for each correspodig term i the left-had product. From 25,, m, so m is also true. Now we oly eed to show that the iterator limits i the two products are also the same. The first term of the left-had product is the smallest multiple of m that is also a multiple of, which is m/, m. This term occurs whe the iterator i 24 is at /, m r from 22. This agrees with the value of the first term at q 1 for the right-had product. The fial value for the iterator q i the product of 24 is From 23, pm, ad p, so that from the GCD defiitio p/r p, m/. 26 p, m. Therefore from 22, r p. The fial term i the right-had product is therefore p/rr p, which, give 23, is also the fial term i the left-had product of 24. Therefore, with the same set of terms i both products of 24, the two sides are equal, ad the proof of Lemma 1 is complete. We cotiue the proof of 13 by showig that coditio 23 i Lemma 1 is satisfied for all upper limits of the three product iterators i the right-had side of 21. This is trivially true for f, ad is true for f give that m, ad therefore it is also true for f. Therefore we ca use 24 from Lemma 1 to replace each product i the right-had side of 21 to give m + /r qr q1 /r qr q1 /r q1 qr r /r /r q r /r /r q1. 27 /r q r /r q q1 q1
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 2008, #A05 8 The powers of r i 27 cacel. Elimiatig the remaiig terms i r from Lemma 1 usig 22, we get m +,m/ q1,m q1 q q,m / q1 q. 28 From Lemma 1, all the limits o the product iterators of 28 are itegers. Therefore we ca defie each product as a factorial to get m +, m!, m/!, m /!. The, m / term is expaded to give m +, m!, m, m/!, m, m/!, m/ The last step is true from the defiitio of the biomial coefficiet. This proves 20 ad completes the proof of the q-biomial root of uity theorem 13 for the case whe m. Case 2: m I this case, if there is oe more zero term i the product umerator of 14 tha i its deomiator, the the q-biomial is zero. This follows because all terms are of a similar form, so that the ratio of ay pair of terms i the umerator ad deomiator, if the latter both ted to zero, is fiite. All other terms i the product are fiite. Therefore ay extra zero term i the umerator product implies a overall result of zero. We first fid the umber of zero terms i the umerator, ad the the umber i the deomiator. Fially, we show that the former exceeds the latter. We ow fid the umber of zeros i the product of the umerator terms i 14, where each umerator term is N m 1 e 2iπ +m/. This fuctio is zero whe For all m,, b Z where > 0, it is trivially true that. m. 29 mb, m. 30
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 2008, #A05 9 Note that the right-had side is also chose to be a iteger multiple of m. Comparig 29 with 30, a zero i the umerator of 14 occurs whe From the product 14, 1, so that 31 implies that 0 b b, m. 31 1, m. 32 For some cases the fractio i 32 is ot iteger, so we fid the ext smaller iteger usig the floor operator: 1, m 0 b. 33 The umber of iteger values of b i the rage of 33 gives the umber of zeros i the umerator of 14 as 1, m Z um 1 +. 34 We ow fid the umber of zeros i the product of the deomiator terms i 14, where each deomiator term is D m 1 e 2iπm/. This fuctio is zero whe m. Hece from 30, D m 0 whe b, m. 35 From the product 14, 1, so that for iterator values 1, 35 gives b, m. 36 From the defiitio of the GCD fuctio itself, 1, m, so that 0 <, m 1. 37 Therefore from 36 ad 37, as b is a iteger, we have b 1. At the other ed of the rage,, so from 35,, m b. 38 As m, the 2, ad ay prime factors of that do ot divide m also do ot divide, m, so that, m. 39
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 2008, #A05 10 From 39, the right-had side of 38 is ot a iteger, so for a iteger b,, m b. 40 Earlier we foud b 1. From 40, we ow have a iclusive rage for b:, m 1 b. The umber of zeros i the deomiator of 14 is the umber of iteger values tae by b which is, m Z deom. 41 To complete the proof, we ow show that Z um > Z deom. Let r, m. 42 For Case 2, m, so that, m, which implies that r. Furthermore, m, ad therefore, m so that, m < ad r 2. For r, Z ad r 2, it is readily apparet that 1 r. 43 r r These costraits allow substitutio of 42 ito 43 to get, m 1, m m. 44 Therefore, from 41 ad 44, 1, m Z deom m. 45 From 34 ad 45, we fially get Z deom Z um 1 m. So there is oe fewer zero term i the deomiator of 14 tha the umerator, ad the case for 13 where m is also proved. This completes the proof.