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( u ; a ) ( v ; b ) ( u ; a ) ( v ; b ) = z − z . ( x ; a ) ( y ; b ) ( x ; a ) ( y ; b ) ( u ; a ) ( v ; b ) (2) ( x + a )( y + b ) z − ( u + a )( v + b ) z ( x ; a ) ( y ; b ) � � � Theorem. Supposethat 0 ≤ m ≤ nandrbenaturalnumbers.Thereholds Thenwehav

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Share "( u ; a ) ( v ; b ) ( u ; a ) ( v ; b ) = z − z . ( x ; a ) ( y ; b ) ( x ; a ) ( y ; b ) ( u ; a ) ( v ; b ) (2) ( x + a )( y + b ) z − ( u + a )( v + b ) z ( x ; a ) ( y ; b ) � � � Theorem. Supposethat 0 ≤ m ≤ nandrbenaturalnumbers.Thereholds Thenwehav"

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(1)

MANY IDENTITIES FROM ONE

CHU WENCHANG

We specialise an ingeniously contrived identity to obtain a plethora of startling combinatorial identities. Limiting cases include well known evaluations of ζ (2), ζ (4) and ζ (6).

Let x , y, u and v be formal parameters and {ai}i≥0, {bj}j ≥0 be two sequences taken from a commutative ring. De�ne the a-rising factorial for n ≥ 0 by

(1) (x ; a)0=1, (x ; a)n=

n

k=1

(x + ak−1) (n > 0).

Then we have the following algebraic identity.

Theorem. Suppose that 0 ≤ m ≤ n and r be natural numbers. There holds

n

k=m

(x + ak)(y + br+k)z − (u + ak)(v + br+k)� (x ; a)k(y; b)r+k

(u; a)k+1(v; b)r+k+1

zk (2)

= (x ; a)n+1(y; b)r+n+1

(u; a)n+1(v; b)r+n+1

zn+1(x ; a)m(y; b)r+m

(u; a)m(v; b)r+m

zm.

Entrato in Redazione l’8 luglio 1996.

AMS Subject Classi�cation (1991): Primary 05A10; Secondary 05A19.

Key Words and Phrases: Binomial coef�cient, Combinatorial identity.

Partially supported by IAMI (Milano, 1994)

(2)

Proof. Notice that the summand on the left of (2) can be split as (x ; a)k+1(y; b)r+k+1

(u; a)k+1(v; b)r+k+1

zk+1(x ; a)k(y; b)r+k

(u; a)k(v; b)r+k

zk.

Then the theorem follows immediately from the diagonal cancellation. � In what follows, we shall demonstrate a number of interesting identities from (2) which could be served as the reason for the present paper’s title.

First of all, take ak =bk = −k, u = −x , v = −y, m = +1 and r = 0 in (2). Then z = ±1 correspond to the following identities.

Proposition 1.

n

k=1

k

x

k

� �y

k

x+k

k

� �y+k

k

� = x y 2(x + y)

� 1 −

x

n+1

� � y

n+1

x+n

n+1

� �y+n

n+1

(3a) ,

n

k=1

(−1)k(x y + k2)

x

k

� �y

k

x+k k

� �y+k k

� = x y

2

−1 + (−1)n

x

n+1

� � y

n+1

x+n

n+1

� �y+n

n+1

(3b) .

Taking y = v = 0 and z = bk =1 in (2) yields Proposition 2.

(4)

n

k=m

(x − u) (x ; a)k

(u; a)k+1 = (x ; a)n+1

(u; a)n+1(x ; a)m (u; a)m. When m = 0, this reduces to [2]

Corollary 3.

(5) 1 +

n

k=1 k

i=1

x + ai−1

u + ai

= u + a0

u − x

 1 −

n

j =0

x + aj

u + aj

 .

This identity contains two formulae of Gould (see [3], Eq.(2.1–2.2) and (4.1–4.2)) as special cases:

n

k=0

(−1)k

x

k

u+k

k

� = u u + x

� 1 −

x

n+1

−u

n+1

(6a) ,

n

k=0

x

k

u

k

� =

u + 1 u − x + 1

 1 −

x

n+1

u+1 n+1

(6b) .

Next, let {ωk} 1≤k≤ p be the p-th roots of unity. The substitution of x → −xp, u → −up, and a → apin (4) gives that

(3)

Proposition 4.

n

k=m p

j =1

(−ωjx ; a)k

(−ωju; a)k+1

(7) =

= 1

xpup

p

j =1

(−ωjx ; a)m

(−ωju; a)m

p

j =1

(−ωjx ; a)n+1

(−ωju; a)n+1

 .

When m = 0, this reduces to an identity due to Chu [2]. In the latter case, setting ak =y − k would generate the following

Proposition 5.

(8)

n

k=0 p

j =1

y−ωjx

k

y−ωju−1 k

� =

ypup xpup

 1 −

p

j =1

y−ω

jx

n+1

y−ωju n+1

 .

For y = −1 and y = u = 0, Eq.(8) corresponding to two interesting special formulas [4] displayed below:

Corollary 6.

n

k=0 p

j =1

k−ωjx k

k−ω

ju+1

k

� =

1 − up xpup

 1 −

p

j =1

n−ωjx+1 n+1

n−ω

ju+1

n+1

(9a) ,

n

k=m

(−1)kp

p

j =1

�ωjx k

=

p

j =1

n − ωjx n

(9b) .

In fact, the last formulae is the limit, as y → 0, of

n

k=0

(−1)kp

p

j =1

y−ωjx k

k−y

k

� = yp xp

 1 −

p

j =1

y−ω

jx n+1

y n+1

which is a reformulation of Eq.(8) under u = 0.

For p = 2r , an even integer, we havej}1≤ j ≤2r = {ωj}1≤ j ≤r

{−ωj}1≤ j ≤r. From the in�nite product

(10) sin π t

πt =

n=1

� 1 − t2

n2

� , we can derive limiting forms of (9a–9b):

(4)

Proposition 7.

k=0 r

j =1

k−ωjx k

� �k+ωjx k

k−ω

ju+1

k

� �k+ω

ju+1

k

� =

1 − u2r x2ru2r

 1 −ur

xr

r

j =1

sin π ωjx sin π ωju

(11a) ,

k=0 r

j =1

�ωjx k

�� −ωjx k

=

r

j =1

sin π ωjx π ωjx . (11b)

These contain the following simple but interesting examples:

Corollary 8.

k=0

1

u−2 k

� �−u−2 k

� = (1 − u

−2)�

1 − πu sin π u

(12a) ,

k=0

x k

�� −x k

= sin π x πx . (12b)

When r = 1, 2, 3, taking the limits as u → 0 and x → 0 successively in (11a), we obtain the celebrated Eulerian formulae:

n>0

n−2= π2/6, (13a)

n>0

n−4= π4/90, (13b)

n>0

n−6= π6/945.

(13c)

Finally, put ak = bk = −k, x → x − t , y → −x − t , u → u − t , v → −u − t and z = 1 in (2). We get the following nice result.

Proposition 9.

n

k=0

x−t

k

� �−x−t

r+k

t −u+k

k

� �t +u+r+k

r+k

(14) � =

= u2t2 (u − x )(u + x + r)

� �−x−t

r

−u−t

r

� −

x−t

n+1

� �−x−t

n+r+1

u−t

n+1

� �−u−t

n+r+1

� .

(5)

After a trival manipulation in the cases of u = t = 0 and x = t − 1 = 0, we have the respective consequences:

Corollary 10.

n

k=0

x k

��

−x r + k

=(−1)r x x + r

x − 1 n

� � −x − 1 r + n

(15a) ,

n

k=0

1

u−2 k

� �−u−2 r+k

(15b) � =

=(−1)r u2−1 u(u + r)

 1

−u−1 r

� −

1

u−1 n+1

� �−u−1 r+n+1

 .

In view of (10), their limiting forms are produced as follows:

Corollary 11.

k=0

x k

��

−x r + k

= sin π x π(x + r), (16a)

k=0

1

u−2 k

� �−u−2 r+k

� =

u2−1 u(u + r)

� 1

u+r

r

� − πu sin π u

(16b) .

For r = 0, these reduce to (12a) and (12b) respectively.

Before closing this short paper, I would like to sketch an application of the theorem to basic hypergeometric series.

Recall that the q-factorial is de�ned by (17) [x ; q]0=1, [x ; q]n =

n

k=1

(1 − x qk−1) (n > 0).

Then the replacement ak =q−kt in (7) gives rise to Proposition 12.

(18)

n

k=0

qkp

p

j =1

jx /t; q]k

[qωju/t; q]k

= tpup xpup

 1 −

p

j =1

jx /t; q]n+1

ju/t; q]n+1

 .

(6)

When t = p = 1, its limiting form reduces to an elegant result [1]:

Corollary 13.

(19)

k=0

qk [x ; q]k

[qu; q]k

= u − 1 u − x

1 − [x ; q]

[u; q]

� .

From the examples listed above, it can be noted that identity (2) in the theorem contains a plentitude of combinatorial identities. The writer believes that it should do more signi�cant ones as its special cases. For the interested readers, there is no harm to try.

REFERENCES

[1] G.E. Andrews, The Theory of Partitions, Addison-Wesley, London, 1976.

[2] W.C. Chu, A family of algebraic identities with their applications, J. of Dalian Institute of Tech.,29:1 (1989), pp. 1-8; MR90k:05021.

[3] H.W. Gould, Combinatorial Identities, Morgantown, 1972.

[4] L.C. Hsu - Y.Z. Ou, A pair of combinatorial identities with their applica- tions, Chinese 1st Conf. on Combinatorial Mathematics, Dalian, 1983.

[5] J. Riordan, Combinatorial Identities, John Wiley & Sons, New York, 1968.

Istituto di Matematica “Guido Castelnuovo”, Universit`a degli Studi di Roma “La Sapienza”, Piazzale Aldo Moro 2, 00185 Roma (ITALY)

References

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