X X PDF 2! + t4. 4! 3 t6.

In Exercises 1-15 use mathematical induction to establish the formula for n ? 1. 1. 1. 2. + 2. 2. + 3. 2. + ··· + n. 2. = n(n + 1)(2n + 1).

MATHEMATICAL INDUCTION SEQUENCES and SERIES

?n is always divisible by 30. Use the fact that n. 5. ?n = (n?1)n(n+1)(n. 2. +1) to prove that it is divisible by 2 and 3 as well as 5.

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Untitled

principle of induction P(n) is true for all positive integers n > 1. herefore each term in the product (1+2+22)(1 +3 +3² +3³)(1 +.

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3 Mathematical Induction 31 First Principle of

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Ann. Sci. Math. QuÂebec23(1999), no. 1, 63±72.

ON Aq-SEQUENCE THAT GENERALIZES

THE MEDIAN GENOCCHI NUMBERS

UO-NIUHAN ANDJIANGZENG

RÂESUMÂE. Dans un article prÂecÂedent [7], nous avons dÂe®ni unq-analogue des nombres de Genocchi mÂedians H 2n+1 . Dans cet article nous dÂemontrons unq- analogue d'unrÂesultat de Barsky [1] sur l'Âetude2-adiquedes nombres de Genocchi mÂedians. A BSTRACT. In a previous paper [7], we de®ned a sequence ofq-median Genocchi numbers H 2n+1 . In the present paper we shall prove aq-analogue of Barsky's theorem about the 2-adic properties of the median Genocchi numbers.

1. Introduction.TheGenocchi numbersG

2n ,n?1, (see [2, 10]), are usually de®ned by their exponential generating function 2 t e t+1 ?t+ Xn? 1 ??1? nG 2n t 2n ?2n?! ?t? t 2 2!+ t 4 4! ?3 t 6 6!+17 t 8 8!

Themedian Genocchi numbersH

2n+1 ,n?0, (see [1, 11]), can be de®ned by H 2n+1 Xi? 0 ??1? iG 2n?2i n 2i+1 ??n? 0?

For exampleH

7 ?3G 6 ?G 4 ?9?1?8. A less classical de®nition of the Genocchi numbers and median Genocchi numbers is the so-called Gandhi generation [3]: G 2n+2 ?Bn ?1??H 2n+1 ?Cn ?1??n?1? whereBn ?x?andCn ?x?,n?1, are polynomialsde®ned by: ?B 1 ?x??x 2 ?B n ?x??x 2 fBn? 1 ?x+1??Bn? 1 ?x?g?n?2? (1) and ?C 1 ?x??1?C n ?x?? ?x +1?f?x+1?Cn? 1 ?x+1??xCn? 1 ?x?g?n?2? (2) RecËu le 29 juillet1997 et, sous forme dÂe®nitive, le 27 fÂevrier 1998. c ?Association mathÂematique du QuÂebec

64On aq-sequence that generalizes the median Genocchi numbers

The reader is referred to [1, 3, 4, 7] for further information about these numbers.

In [7]

1 , we introduced aq-analogue of the above polynomialsby letting ?C 1 ?x? q??1?C n ?x? q?? ?

1+qx??q

?xCn? 1 ?x? q???n?2(3) where ?q f?x?? ?f?1+qx??f?x????1+qx?x?for any polynomialf?x?.As Cn ?x?1??Cn ?x?forn?1, the polynomialsCn ?1?q?,n?1, de®ne aq-analogue of the median Genocchi numbers H 2n+1 In [7], we proved somecombinatorialinterpretationsand analyticalproperties of the polynomial Cn ?1?q?. In this paper, we investigate divisibility properties ofCn ?1?q?.

Our resultsmay be seen as

q-analogues ofthe2-adicpropertiesof themedianGenocchi numbers derived by Barsky [1]. We ®rst recall his result. It will be reproved in section

2 using a different technique.

Theorem 1.(Barsky)For

n?2, the n-th median Genocchi numberH 2n+1 is divisible by2 n?1 and H 2n+1 2 n?1

2?mod 4??ifnis odd?

3?mod 4??ifnis even?

Ourq-analogue of this result is the following theorem.

Theorem 2.The

q-polynomialsC 2n+1 ?1?q?,n?1, andC 2n+2 ?1?q?,n?0,are divisible by ?1+q? 2n+1 . Moreover, forn?1,let c 2n+1 ?q?? C 2n+1 ?1?q? ?1+q? 2n+1 ?c 2n ?q?? C 2n 1?q? ?1+q? 2n?1 ?(4) Then c 2n

1????1?

n +1 G 2n ?c2n+1 ??1?? ??1? n +1 G 2n+2 ?n?1?(5)

The ®rst values of the sequence

fcn ?q?gare the following: c 2 ?q??c 3 ?q??1?c 4 ?q??1+3q+2q 2 +q 3 ?c 5 ?q??1+5q+5q 2 +5q 3 +2q 4 +q 5 We also calculate the ®rst values of the sequencesfH 2n+1 g,fcn ?1?gandfcn ??1?gin the following table: n23 4 5 6 7 8 H 2n+1

2 8 56 608 9440 198272 5410688

cn ?1?1 1 7 19 295 1549 42271 cn ??1?1 1 -1 -3 3 17 -17 1 Preprint available athttp://cartan.u-strasbg.fr/?guoniu/papers

G. Han and J. Zeng65

Note thatH

2n+1 ?2 nc n ?1?or 2 n?1 cn ?1?according to the parity ofn.

Throughoutthispaper,the

q-analogueofnisdenotedby?n?q ?1+q+q 2 +???+q n? 1

Notice that

?n?1 ?n,?

2n+1??

1 ?1and?2n?q

1+q???n?q

2. Flajolet [6] has shown how to derive certain arithmetic properties of a sequence from the continued fraction expansion of its ordinary generating function. Following Flajolet's idea we derive Theorem 1 from the known continued fraction expansion of the median Genocchi numbers (see section 2). Theorem 2 will be proved by a different method (see Section 3). In Section 4, we shall derive some divisibility properties of cn ?q?by continued fraction manipulations that will bring more light in Theorem 2. Finally we make some comments on a possible combinatorial approach to Theorem 2 following a recent work by Kreweras [8] (see Section 5).

2. A new proof of Theorem 1.In thispaper, the continued fractions are considered as

formal power series. Clearly, Cn ?1?q??Cn +1?

0?q?forn?1, so that

1+ Xn? 1 Cn ?1?q?t n? Xn? 1Cn ?0?q?t n? 1 ?(6) In our previous paper [7, Corollary 4], we proved the following identity : Xn?1 Cn ?x? q?t n? t 1?

1+qx??t

1? q?

1+qx??t

1? 2 ?q ???2?q+q 2 x??t 1 q?2?q ???2?q+q 2 x??t ?(7)

It follows from (6) and (7) that

Xn?1Cn

?1?q?t n?1+ t 1? 1?q ??2?q ?t 1? q? 1?q ??2?q ?t 1? 2?q ??3?q ?t 1? q? 2?q ??3?q ?t 1? 3?q ??4?q ?t 1? q? 3?q ??4?q ?t ...(8) 1 100
1? 2 q ?t 1? q? 1? 2 q ?t 1? 2? 2 q ?t 1? q? 2? 2 q ?t 100
3? 2 q ?t 100
q? 3?quotesdbs_dbs35.pdfusesText_40

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How to prove n is true for all integers n 1?

Does f(n) = n 1+n2 converge?

Is p divisible by Q over C?

ON Aq-SEQUENCE THAT GENERALIZES

THE MEDIAN GENOCCHI NUMBERS

UO-NIUHAN ANDJIANGZENG

1. Introduction.TheGenocchi numbersG

Themedian Genocchi numbersH

For exampleH

64On aq-sequence that generalizes the median Genocchi numbers

In [7]

1+qx??q

Our resultsmay be seen as

2 using a different technique.

Theorem 1.(Barsky)For

2?mod 4??ifnis odd?

3?mod 4??ifnis even?

Theorem 2.The

1????1?

The ®rst values of the sequence

2 8 56 608 9440 198272 5410688

G. Han and J. Zeng65

Note thatH

Throughoutthispaper,the

Notice that

2n+1??

1+q???n?q

2. A new proof of Theorem 1.In thispaper, the continued fractions are considered as

0?q?forn?1, so that

1+qx??t

1+qx??t

It follows from (6) and (7) that

Xn?1Cn