Super congruences concerning Bernoulli polynomials. Zhi-Hong Sun

Transcription

1 It J Numer Theory 05, o8, Super cogrueces cocerig Beroulli polyomials Zhi-Hog Su School of Mathematical Scieces Huaiyi Normal Uiversity Huaia, Jiagsu 00, PR Chia zhihogsu@yahoocom Let p > e a prime, ad let a e a ratioal p-adic iteger Let {B x} deote the Beroulli polyomials give y B 0, 0 B 0 ad B x 0 B x 0 I this paper, usig Beroulli polyomials we estalish cogrueces for a a mod p ad a mod p As a cosequece we solve the followig cojecture of ZW Su: where H H p B p mod p, Keywords: Cogruece; Beroulli polyomial; Beroulli umer Mathematics Suject Classificatio 00: Primary A07, Secodary B8, 05A9 Itroductio The Beroulli umers {B } ad Beroulli polyomials {B x} are defied y B 0, B 0 ad B x 0 0 B x 0 The Euler umers {E } are defied y E 0 ad E [/] E, where [a] is the greatest iteger ot exceedig a I [S5] the author itroduced the sequece {U } give y U 0 ad U [/] U It is well ow that B + 0 ad E U 0 for ay positive iteger {B }, {E } ad {U } are importat sequeces ad they have may iterestig properties ad applicatios See [B], [MOS] ad [S-S]

2 Let Z e the set of itegers For a prime p let Z p deote the set of ratioal p adic itegers For a p adic iteger a let {0,,, p } e give y a mod p I [S7] the author showed that for ay odd prime p ad a Z p, a a p mod p 0 As poited out i [S7], we have, , 5, 7 4 Let p > e a prime ad a Z p I [T] Tauraso otaied a cogruece for I [Su], the author s rother ZW Su cojectured that 4 4 H p B p mod p, where H Let p > e a prime, a Z p ad t a /p I this paper we prove that a a p tt + B p a B p a B p mod p p p As cosequeces we completely determie 7, 4 4, I particular, we cofirm ad prove that 4 7 where q p a a /p We also show that ad completely determie /4 5 where is the Legedre symol p 4 mod p q p pq p + p q p B p mod p, a B p a B p p p a B p a mod p a a a mod p i the cases a,, For example, 4 q p + p q p + E p mod p, p mod p

3 Cogrueces for a For ay positive iteger ad variale a let S a a a a mod p The S a S a { a a a a a a + a a a a a } a a By [S7, 45] or iductio o, a a + a a a a 0 Thus, S a S a a a a a Lemma [S7, Lemma 4] Let p e a odd prime, m {,,, p } ad t Z p The m + pt pt p m p t m + p t m H m mod p By ad Lemma, if p is a odd prime ad a Z p with a 0 mod p, the S a S a tt + p a p a tt + p p a mod p, where t a /p Lemma Let p > e a prime ad t Z p The pt pt p tb p mod p Proof For {,,, p } we see that pt pt ptpt pt + pt pt pt! ptpt + p t p t!

4 Thus, pt ptpt + p t pt mod p pt p t pt mod p By [L] or [S, Corollary 5], 0 mod p ad pb p mod p Thus the result follows Lemma Let p > e a prime, a Z p, a 0 mod p ad t a /p The a a p p tb p r r + pt r r + p p t r r mod p Proof For we have a + p + ad so a + a + p +pt Usig we see that Note that S a S a p tt + p S a + S a a + p p tt + p p tt + p r + p r p r r + pt mod p + + pt a + We the otai p r r + pt r ptr + p t r pt r p r r pt r p r r + p p t r r ptr + p t r r mod p p S a S a r r + pt r r + p tp r r mod p By Lemma, S a S pt p tb p mod p Thus the result follows Theorem Let p > e a prime, a Z p, a 0 mod p ad t a /p The a a p tt + B p a B p a B p p p 4 mod p

5 Proof It is well ow that see [MOS] m r B +m + B +, + r 4 B x + y y B x, B x B x 0 Thus, usig Euler s theorem we see that pt r p r r r p pt r p r r r p pt B p + B p p p pt B p p p pt B p pt a p p pt p p p p p p 0 B p + B p p p B p B p p p B p pt a B p a + B p a B p p p p p pt B p a p p pt B p a B p a B p p p mod p By [S, Lemma ], B m a Z p for m 0 mod p ad pb m a Z p for m 0 mod p Thus, pt r p r + B a B p p p p r r pt B p a + p p ptb p a ptb p a p p pt B p a p t B p a + p t B p a p t B p +p a mod p By [S, Corollary ], B p +p a p p + p B p a p B p a mod p 5

6 Thus, 5 pt r p r B a B p p p p p t B p a mod p r r By [S, Lemma ], p r r p r r p 4 B p a B p p Thus, from Lemma, 5 ad we derive that a a p p tb p r Bp a B p mod p r + pt r r + p tp r p tb p B p a B p p p p t B p a p t B p a B p r p tt + B p a B p a B p p p mod p This completes the proof Lemma 4 [MOS] For ay positive iteger we have i ii iii B B, B B B 4 4, B Theorem Let p > e a prime The Proof By Lemma 4, B, B q p pq p + p q p B p mod p, q p pq p + p q p + 7 B p mod p, 4 4q p + q p p q p + q p B p + p 4 q p + q p B p p p B p B p mod p, mod p

7 By [S4, p87], B p p B 4 4 p p 8B p mod p, B p p p B p p 9B p mod p B p B p p p B p B p 4 p p B p B p p p qp pq p + p q p mod p, q p pq p + p q p mod p, q p pq p + p q p + q p pq p + p q p mod p Now taig a,, 4 the result i Theorem ad the applyig ad the aove we deduce Remar Let p > e a prime I [T] Tauraso showed that H mod p, which ca e deduced from Theorem with a ad the cogruece [S, Theorem 5c] 0 H q p + pq p p q p 7 p B p mod p By 0, Theorem ii is equivalet to ZW Su s cojecture Cogrueces for a mod p For give positive iteger ad variales a ad with {0,,, } defie a f a, The f a, f a, a a By Lerch s theorem [B, p8] or iductio o, a + + a 0 { a a } a a 7

8 Thus, f a, f a, a { + + a + a + a + a + a a } Lemma Let p e a odd prime, {,,, }, a, Z p, p ad a + pt The Proof By, a pt + pt p r p r r + rr r + r + pt r mod p p f a, f a, f a +, f a, p p + a + a + a + + Sustitutig with + r i the aove we otai f a, f pt, p r + r pt p + r p + + a r + r pt r + pt For r we see that r + pt r + ptr + pt + ptptpt pt r! r! pt r r! r pt! r mod p r ad so f a, f pt, p r p r + r pt p r + r + pt pt p + r 8 r r + r pt r + rr r pt r r mod p

9 O the other had, f pt, pt pt pt pt pt mod p Thus, the result follows Theorem Let p > e a prime, a Z p ad a 0 mod p The a B p a B p p p Proof Set a + pt By [S, Lemma ], a B p a mod p 4 p r p p B p a B p p r B p a mod p As, taig ad p i Lemma ad the applyig 5, 4 ad the ow fact see [S, Corollary 5] a a pt B p B p a p p 0 mod p we see that p r + pt r p r r + pt p r + pt B p a mod p This yields the result Remar Let p > e a prime Taig a i Theorem ad the applyig Lemma 4 we deduce that / q p + pq p mod p I [T], usig a special method Tauraso proved the followig stroger cogruece: / Theorem Let p > e a prime The H /4 q p + p q p + mod p r E p mod p p Proof Taig a i Theorem we see that 4 /4 B p B 4 p 4 4 p B p p mod p p 4 9

10 By 8, B p B p 4 q p p pq p mod p It is ow see for example [S4, Lemma 5] that E 4 + B + 4 Thus, E + p 4 p B p 4 B p 4 mod p Now, p 8 from the aove we deduce the result Theorem Let p > e a prime The / Proof Taig a / q p + p 4 q p + i Theorem we see that B p B p p p p U p mod p p B p mod p By 7, B p B p q p p pq 4 p mod p By [S5, p7], B p U p mod p Now, from the aove we deduce the result Theorem 4 Let p > e a prime The / q p q p + p q p + 4 q p + 5 U p mod p p Proof Taig a i Theorem ad the applyig 9 ad the fact see [S5, p] B p 0U p mod p we deduce the result Acowledgmet The author is supported y the Natioal Natural Sciece Foudatio of Chia grat No 7 Refereces [B] H Batema, Higher Trascedetal Fuctios, VolI, McGraw-Hill, New Yor, 95 [L] E Lehmer, O cogrueces ivolvig Beroulli umers ad the quotiets of Fermat ad Wilso, A of Math [MOS] W Magus, F Oerhettiger ad RP Soi, Formulas ad Theorems for the Special Fuctios of Mathematical Physics rd ed Spriger, New Yor, 9, pp 5- [S] ZH Su, Cogrueces for Beroulli umers ad Beroulli polyomials, Discrete Math

11 [S] ZH Su, Cogrueces cocerig Beroulli umers ad Beroulli polyomials, Discrete Appl Math [S] ZH Su, Cogrueces ivolvig Beroulli polyomials, Discrete Math [S4] ZH Su, Cogrueces ivolvig Beroulli ad Euler umers, J Numer Theory [S5] Z H Su, Idetities ad cogrueces for a ew sequece, It J Numer Theory [S] [S7] Z H Su, Some properties of a sequece aalogous to Euler umers, Bull Aust Math Soc ZH Su, Geeralized Legedre polyomials ad related supercogrueces, J Numer Theory [Su] ZW Su, p-adic cogrueces motivated y series, J Numer Theory [T] R Tauraso, Cogrueces ivolvig alteratig multiple harmoic sums, Electro J Comi 700 #R, pp [T] R Tauraso, Supercogrueces for a trucated hypergeometric series, Itegers 0, #A45, pp