GENERALIZED LEGENDRE POLYNOMIALS AND RELATED SUPERCONGRUENCES

Transcription

1 J. Nuber Theory 0, o., 9-9. GENERALIZED LEGENDRE POLYNOMIALS AND RELATED SUPERCONGRUENCES Zhi-Hog Su School of Matheatical Scieces, Huaiyi Noral Uiversity, Huaia, Jiagsu 00, PR Chia Eail: Hoeage: htt:// Abstract. For ay ositive iteger È ad variables a ad x we defie the geeralized Legedre a olyoial P a, x by P a, x a 0 x. Let be a odd rie. I this aer we rove ay cogrueces odulo related to P a, x. For exale, we show that P a, x a P a, x od, where a is a ratioal adic iteger ad a is the least oegative residue of a odulo. We also geeralize soe cogrueces of Zhi-Wei b a od. Su, ad establish cogrueces for È 0 Æ 5 ad È 0 MSC: Priary A07, Secodary C5, 05A0, 05A9, E5 Keywords: Cogruece; bioial coefficet; geeralized Legedre olyoial a. Itroductio. Let be a oegative iteger ad let [ ] be the greatest iteger fuctio. The the faous Legedre olyoial P x is give by. P x [/] 0 x! d dx x see for exale [B,.79-80]. For ay ositive iteger ad variables a ad x we itroduce the geeralized Legedre olyoial. P a x a a x 0 a + x. 0 0 a + x We ote that a a+ ad a a+ a+ a x ad P x P x see [B,.80].. Clearly P a x P The author is suorted by the Natioal Natural Sciece Foudatio of Chia grat o. 7.

2 Let > be a rie. I 00, based o his wor cocerig hyergeoetric fuctios ad Calabi-Yau aifolds, Rodriguez-Villegas [RV] cojectured the followig cogrueces: od 7 od od od where is the Legedre sybol. These cogrueces were later cofired by Morteso [M-M] via the Gross-Koblitz forula. Recetly the author s brother Zhi-Wei Su [Su] osed ore cojectures cocerig the followig sus odulo : 0 x 0 7 x 0 x 0 x. For the rogress o these cojectures see [S-S5]. As observed by Tauraso [T], Zudili ad the author [S,.9-97, 90], [S,.95], [S5,.8], we have 5 7. This is the otivatio that we itroduce ad study P a x od. Let Z be the set of itegers. For a rie let Z deote the set of ratioal adic itegers. For a adic iteger a let a {0... } be give by a a od. Let be a odd rie ad a Z. I this aer we show that.7 P a x a P a x od. Note that P a. Taig x i.7 we obtai.8 0 a P a a od.

3 For a we get.-. iediately fro.8. If a is odd, by.7 we have a a.9 P a 0 0 od. 0 This geeralizes revious secial results i [S] ad [Su]. If f0 f... f are adic itegers, we rove the followig ore geeral cogruece: a a.0 a f f 0 od ca be viewed as vast geeralizatios of soe cogrueces roved i [S,S] with a ad [Su,Su] with a. Whe a od, taig f / i.0 we get. 0 a 0 0 od. This ilies several cojectures of Rodriguez-Villegas [RV], see I this aer we also establish the cogruece a + P a + x a + xp a x + ap a x 0 od for a 0 od ad use it to rove our ai results. As a alicatio, we deduce the cogruece for / 0 5 od >, see Theore.. I Sectio, we obtai a geeral cogruece for b a 0 od, where is a odd rie ad a b Z.. Geeral cogrueces for P a x od. Lea.. Let be a ositive iteger. The Proof. It is clear that a + P a + x a + xp a x + ap a x a a + x +. a + aa + a + a + a + + a + a + a + a + +. Thus, a + a + + a + a + a + a + a + a + a + a + a aa + a! + a + a + + a + a + a + a

4 Therefore, a + a + + a + a + a + a +. a + a +. For ay egative iteger set α 0. Usig. we deduce that a + P a + x a + xp a x + ap a x { a + + a + a + + a 0 a + + x a + }x a + x + a + { a + + a + + a + + a 0 a + a + }x a + + a a + x +. a + This roves the lea. Theore.. Let be a odd rie ad a Z. The a + P a + x a + xp a x + ap a x a + a + a a + a a x od if a 0 od, a + a + x od if a 0 od, a + a x od if a od. Proof. Clearly a a + If a 0, the aa a a + a + a +!. aa a a + a + a + a a a + a { a a a + a + a + a } a a + a a! a a + a a + a a a + od.

5 If a 0, the a + If a, the a 0 od ad so a a + aa + a a a! aa + aa +! aa + + aa + + a + a + a + a od. a a + a + a a + + a + od. Now uttig all the above together with Lea. i the case we deduce the result. Lea.. Let be a odd rie ad let t ad x be -adic itegers. The + x x P t x t + t + t od ad so P t x od. Proof. It is clear that P t x 0 t + t t + x t t + x x { + t + t + x x } { + x x } t + t + od. Sice ±x ±x od, by the above we obtai P t x od. This roves the lea. Lea.. Let be a odd rie ad let t ad x be adic itegers. The P + t x tx + t x { + t + x + x x + x + x + x + } od 5

6 ad so P + t x x od. Proof. It is clear that ad so P + t x t + t + + x 0 + t + t + x + t + x + t + t x P + t x + t x + t + + t t t t + + x + x od + t + t x + x x x + t + t + + t + t x { + t + x + x x x + x x x } tx + t x { + t + x + x x x + x + x + } + od. Observe that ±x ±x od. Fro the above we see that { x P + t x tx + t + x x tx + tx x od. This coletes the roof. + + x Theore.. Let be a odd rie ad a Z. The + x P a x a P a x od x } x

7 ad so 0 a x a x 0 od. Proof. Suose {... } ad t Z. Fro Theore. we have Thus,. + + tp + + t ±x + t + ±xp + t ±x + + tp + t ±x 0 od. + + tp + + t x + P + + t x + t + xp + t x P + t x + tp + t x P + t x od. Fro Lea. we ow that Fro Lea. we see that Thus, P t x P t x od. P + t x + P + t x x t + x { x + t + + x x } x x t + t + 0 od. + x + x t + t + x + x x + P + t x P + t x 0 od for 0. By. ad iductio we deduce that P +t x P +t x 0 od for all Sice a a + t for soe t Z, we see that P a x a P a x od ad hece 0 a This coletes the roof. x P a x a P a x a 0 a x od. Rear. I the case a, Theore. was give by the author i [S] ad ideedetly by Tauraso i [T]. I the cases a, Theore. was give by Z. W. Su i [Su]. 7

8 Corollary.. Let be a odd rie ad a Z. The a a a od. 0 Proof. Taig x i Theore. we obtai the result. As etioed i Sectio, taig a i Corollary. we deduce.-.. Corollary.. Let be a odd rie ad a Z with a od. The a a 0 od Proof. Taig x i Theore. we obtai the result. Puttig a i Corollary. we deduce the followig cogrueces: od for od 5 0 od for od 8 0 od for 5 7 od od for od where is a rie greater tha. We rear that. was cojectured by Z. W. Su ad roved by the author i [S] ad Tauraso i [T], ad. was cojectured by the author i [S] ad roved by Z. W. Su i [Su]..5 ad. were cojectured by Z. W. Su ad fially roved by hi i [Su], although the author roved the corresodig cogrueces odulo earlier. Lea.. Let be a ositive iteger. The a a + 0 a + Proof. Set a f ad g + It is easily see that g g f. Thus, f f0 + 0 This roves the lea.. a + g g f0 g0 + g g. 8.

9 Theore.. Let be a odd rie ad a Z. i If {... }, the a x + a x 0 od. ii If a 0 od, the 0 a x + + a x od. Proof. By Theore., 0 a x a x fx where fx is a olyoial of x with ratioal -itegral coefficiets ad degree at ost. Sice d x dx! x ad d dx x! x! x for, we see that a x + a x! d dx fx. As! for ad d dx fx is a olyoial of x with ratioal -itegral coefficiets ad degree at ost, we deduce the first art. Now we suose a 0 od. It is easy to see that a Thus, by Theore. we have a 0 od ad so 0 a a a 0 od. x a x gx where gx is a olyoial of x with ratioal -itegral coefficiets ad degree at ost. It is easy to see that x gtdt 0 0 a x + + a x

10 Thus, usig Lea. we get a a x + + a x a a a a a a od. + 0 Sice a 0 od, we see that a a a a 0 od. a a Thus the secod art follows ad the roof is colete. Corollary.. Let be a odd rie, a Z ad {... }. The a a + a a a od. Proof. Taig x i Theore.i we obtai the result. Corollary.. Let be a odd rie ad a Z with a {... }. The a a + 0 od. 0 Proof. Taig x i Theore.ii we obtai the result. Theore.. Let be a odd rie ad a Z. If f0 f... f are adic itegers, the a a a f f 0 od. 0 0 Proof. Fro Corollaries. ad. we see that a a f 0 f a f + a a a a 0 a f od. This yields the result. Rear. I the case a, Theore. was obtaied by the author i 00. I the cases a, Theore. was recetly obtaied by Z.W. Su i [Su]. 0

11 Theore.5. Let be a odd rie ad a Z with a od. The 0 a 0 od. Proof. Set f /. Fro [S, Exale 0] we ow that 0 f f. Thus, alyig Theore. we deduce the result. Puttig a i Theore.5 we deduce that for ay rie >, od for od 08 0 od for 5 od 0 od for 5 7 od od for od. Here.7 was cojectured by Beuers [Beu] i 987 ad roved by va Hae [vh] were cojectured by Rodriguez-Villegas [RV] ad roved by Z. W. Su [Su].. Cogrueces for P a 0 od. For give ositive iteger ad rie we defie H 0 0 H ad q a a. Theore.. Let be a odd rie ad t Z. i If {0... }, the P + + t 0 0 od. ii If {0... }, the P + t th t + H tq od. Proof. Puttig x 0 i Theore. we see that P a + 0 a a + P a 0 od for a 0 od.

12 Assue {... }. The By Lea., P + + t 0 + t + + t P + t 0 + t + t + + t + t + t + t P + t 0 od. P + t 0 t + t t + t 0 od. Thus, fro the above we deduce that P ++t 0 0 od for This roves i. Now let us cosider ii. Assue {... }. The P + t 0 By [S, Lea.],. Fro Lea. we have + t P + t 0 + t + t + t + t + t + t + P t 0 t + + t + t + t + t + t + t P t 0 P t 0 t P t 0 + th H P t 0 od. + H H od. t t + t q tq od. Therefore, for 0... we have P + t 0 + H H + th H tq + + th t + H tq od. This roves ii ad hece the roof is colete.

13 Corollary.. Let be a odd rie ad let a be a -adic iteger with a 0 od. The P a 0 0 od or P a 0 0 od. Proof. If a + od with {0... }, by Theore.i we have P a 0 0 od. If a od for soe {... }, the a + od ad so P a 0 0 od by Theore.i. For a b Z ot both zero let a b be the greatest coo divisor of a ad b. Theore.. Let be a odd rie, Z, >, r {± ±... ± }, r ad + r. i If r, the r P 0 r/ r/ 0 / ii If ad r, the + s H + s H + s q od if r s od for soe s {... } ad s +r/, 0 od if r s od for soe s {... }. r P 0 r/ r/ 0 / iii If ad r, the + s H + s H + s if r ad s+r/ 0 od if r r P 0 r/ r/ 0 / + s q od s od for soe s {... }, s od for soe s {... }. H + s H + s q od if r s od for soe s {... } ad s+r, 0 od if r s od for soe s {... }. Proof. We first cosider i. If r ad r s od for soe s {... s r }, settig ad t s we fid {0... } ad

14 ++t r. Thus, fro Theore.i we obtai P r 0 P r 0 P ++t 0 0 od. If r ad r s od with s {... }, settig s +r/ ad t s we fid {... } ad + t r. Thus, fro Theore.ii we deduce that r P 0 P r 0 P + t 0 / + s H + s H + s q od. This roves i. Now we cosider ii. Suose ad r. If r s od for soe s {... s r/ }, settig ad t s we fid that {0... } ad ++t r. Thus, fro Theore.i we obtai P r 0 P r 0 P ++t 0 0 od. If r s od for soe s {... }, settig s+r/ ad t s we fid that {0... } ad + t r. Thus, fro Theore.ii we deduce the result. Let us cosider iii. Assue ad r. If r s od for soe s {... }, settig s r ad t s we fid that {0... } ad ++t r. Thus, fro Theore.i we obtai P r 0 P r 0 P + + t 0 0 od. If r s od for soe s {... }, settig s+r ad t s we fid that {0... } ad + t r. Now alyig Theore.ii we deduce the result. The roof is ow colete. Fro Corollary. or Theore. we deduce the followig result. Theore.. Let be a odd rie. The P / 0 0 od P / 0 0 od P / 0 0 od P /5 0 0 od P / 0 0 od P /7 0 0 od P /8 0 0 od P /9 0 0 od P /0 0 0 od P / 0 0 od for od for od for od 8 for od 5 for od for 5 od 7 for 7 od for od 9 for 7 9 od 0 for od P / 0 0 od for 5 7 od.

15 Lea. [L]. Let be a odd rie. The i H q od, H [ ] q od. ii For > we have H [ ] q od ad H [ ] q q od. Theore.. Let > be a rie. The 0 5 { A A od if, A + B ad A, 0 od if. Proof. Fro Theore.i we see that 5 P 0 0 { + H + H + q od if od, 0 od if od. Now we assue od ad A + B with A B Z ad A od. Fro Lea. we have H q od ad Thus, H H H + H By [BEW, Theore 9..], Therefore, P 0 A A This coletes the roof. / / q od. + q q q + q q q od. A A q + q od. + H A A od. + H q q + q + q q Rear. I [S] the author cojectured Theore. ad roved the cogruece odulo. I [Su], Z.W. Su roved the result for od. 5

16 . Cogrueces for b a 0 od. Let be a oegative iteger. For two variables a ad b we defie a b a. S a b. Set a b a Fa It is easy to see that Thus, a b 0 ad Ga a b + b a a bfa + a + Fa + Ga + Ga. Fa + a + 0 Fa + 0 Ga + Ga Ga + Ga 0 Ga +. 0 That is, a b a. a bs a b + a + S a + b a b +. Lea.. Let be a odd rie ad b c t Z. The b t b + ct + t od. 0 Proof. Clearly t b + ct t t t b + ct + b b + t b + t b + t 0 0 x b x 0 This roves the lea. b + t t x dx + t 0 dx + t b u du + t 0 b b 0 b 0 b u b u du + od.. x dx

17 Lea.. Let be a odd rie, {... } ad t Z. The + t t t + t od. Proof. For < we see that + t + t + t + t tt t! t t t t t! t! t! t! t H H + od. For > we also have + t + t + t + t tt t! t t t t + t +! + t t!!! t! t H H + od. Sice! H od, by the above we get + t! t t! H th H t + H t th H t t t + th H H od. To see the result, we ote that H H r + r 0 od. 7 + / r r + r

18 Lea.. Let be a odd rie, {... } ad b t Z. The od b + t if b +, b b t b+ od if b, b b t+ b+ od if > b +. Proof. If b +, settig b r + we fid r Z ad so b t + r t + r t + r t! + r t + r t + od. Now we assue b + ad b b +r. The r Z, b ad b + t! 0 r t b + b + r t r t! r t b + b b t b + b 0 b od. Sice we see that { b if b, b + b if > b +, { b b b b if b, b b b + b b b if > b +. Now cobiig all the above we obtai the result. Theore.. Let be a odd rie ad a b Z. The 0 b a a b a b a b b b od if a > b, b a + b b H b a a H a b a b a H b a od if a b. Proof. If a 0, the result follows fro Lea. with t a ad c. Fro ow o we assue a. Set a a + t ad b b + s. The s t Z. For 8

19 {... }, by. ad Leas.-. we obtai. + ts + t b + + t b S + t b + t b + t + t b b t b b t b+ od if b, b t + t t + H od if b +, b t b b t+ b+ od if > b +. Hece, if a b, the S a b S a + t b a + t b S a + t b a + t + b a + s t S a + t b a + t + b a + s t a + t a + b + s t + t + b a + s t S a + t b a + t S t b od. Now alyig Lea. we see that for a b, S a b b b b a + + s th b H b a + th b a! + th a b + s th b H b a th a + th b a b + sh b th a s th b a a b + b b H b a a H a b a b a H b a od. a Now we assue a > b. Clearly b + t b b t b b t S b + t b + od. This together with. yields b + + ts b + + t b t b + t bs b + t b + b + b b + b b t + t b b od. 9

20 Thus,. S b + + t b b b b + + t b b b + od. This shows that the result is true i the case a b +. Now fro.-. we deduce that for a > b +, b + S a b S a + t b a + t S a + t b b + a + t b + a + t S a + t b a b + This coletes the roof. a b + b + + t S b + + t b b + b b b + a b a b a b b b od. Rear. Let be a odd rie ad a Z. Taig b i Theore. ad the alyig the fact H a H a H a od we deduce Corollary.. I [Su], Z.W. Su showed that 0 a a od. a This ca be deduced fro Theore. by taig b a. Corollary.. Let be a odd rie ad a b Z. The 0 b a b od. a Corollary.. Let be a odd rie ad a Z. The 0 { a 0 od if a 0 od, od if a 0 od. Proof. Taig b 0 i Theore. we deduce the result. 0

21 Corollary.. Let be a odd rie ad a Z. The.5. 0 a 0 od if a a 0 od, + a od if a 0 od, a od if a od. Proof. Taig b i Theore. we deduce the result. Usig the ethod i the roof of Lea. we ca show that for ay ositive iteger, a + a a a a a a 0 0 Theore.. Let be a odd rie. The 0 a + + od if od, x x od if x + y od ad x. Proof. Set a ad b. The. a { if od, if od, ad b { if od, if od. If od, the a > b. Thus, by Theore. we have Sice ! +! + od + by the above we obtai the result i the case od. + + / + / od.

22 Now we assue od ad so x + y with x y Z ad x od. By the roof of Lea. ad Lea.i we have H H q od ad q od. Now alyig the above ad Theore. we deduce that H 0 By [BEW, Theore 9..] we have Hece H + H + H q + q + q od. x x q od. 0 x od. x This roves the result i the case od. The roof is ow colete. Theore.. Let > be a rie. The 0 Proof. Set a ad b. The 5 od if od, A A od if A + B od ad A. b ad a { if od, 5 if od. For od we have a > b. Thus, by Theore. we get 0 Note that od. od.

23 We the get the result i the case od. Now we assue od ad so A + B od with A. By Theore. ad Lea. we obtai 0 By [BEW, Theore 9..], Therefore, 0 H + H + H q + q q + + q q od. A A q q od. q A A q q A A od. This roves the result i the case od. Hece the theore is roved. Refereces [RV] [B] H. Batea, Higher Trascedetal Fuctios, Vol.II, McGraw-Hill, New Yor, 95. [BEW] B.C. Berdt, R.J. Evas ad K.S. Willias, Gauss ad Jacobi Sus, Wiley, New Yor, 998. [Beu] F. Beuers, Aother cogruece for the Aéry ubers, J. Nuber Theory 5 987, 0-0. [vh] L. va Hae, Soe cojectures cocerig artial sus of geeralized hyergeoetric series, i: -adic Fuctioal Aalysis, 99, i: Lect. Notes Pure Al. Math., vol. 9, Deer, New Yor, 997,. -. [L] E. Leher, O cogrueces ivolvig Beroulli ubers ad the quotiets of Ferat ad Wilso, A. of Math. 9 98, [M] E. Morteso, A suercogruece cojecture of Rodriguez-Villegas for a certai trucated hyergeoetric fuctio, J. Nuber Theory 99 00, 9-7. [M] E. Morteso, Suercogrueces betwee trucated F hyergeoetric fuctios ad their Gaussia aalogs, Tras. Aer. Math. Soc , F. Rodriguez-Villegas, Hyergeoetric failies of Calabi-Yau aifolds, i: Norio Yui, Jaes D. Lewis Eds., Calabi-Yau Varieties ad Mirror Syetry, Toroto, ON, 00, i: Fields Ist. Cou., vol. 8, Aer. Math. Soc., Providece, RI, 00,.-. [S] Z.H. Su, Ivariat sequeces uder bioial trasforatio, Fiboacci Quart. 9 00, -. [S] Z.H. Su, Cogrueces cocerig Legedre olyoials, Proc. Aer. Math. Soc. 9 0, [S] Z.H. Su, Cogrueces ivolvig, J. Nuber Theory 0, [S] Z.H. Su, Cogrueces cocerig Legedre olyoials II, J. Nuber Theory 0, [S5] Z.H. Su, Legedre olyoials ad suercogrueces, Acta Arith. 59 0, 9-00.

24 [Su] Z.W. Su, Oe cojectures o cogrueces, arxiv:09.55v59, 0. [Su] Z.W. Su, O sus of Aéry olyoials ad related cogrueces, J. Nuber Theory 0, [Su] Z.W. Su, O sus ivolvig roducts of three bioial coefficiets, Acta Arith. 5 0, -. [Su] Z.W. Su, Suercogrueces ivolvig roducts of two bioial coefficiets, Fiite Fields Al. 0, -. [T] R. Tauraso, A eleetary roof of a Rodriguez-Villegas suercogruece, arxiv:09., 009. [T] R. Tauraso, Suercogrueces for a trucated hyergeoetric series, Itegers 0, A5,.