Matematicko-fyzikálny časopis

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Matematicko-fyzikálny časopis

Pavel Bartoš; Josef Kaucký

Additional Note to our Paper 'A Genesis for Combinatorial Identities'

Matematicko-fyzikálny časopis, Vol. 16 (1966), No. 3, 282--284 Persistent URL:http://dml.cz/dmlcz/126620

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MATEMATICKO-FYZIKALNY ČASOPIS SAV, Ifi. 3, 1006

ADDITIONAL NOTE TO OUR PAPER

„A GENESIS FOR COMBINATORIAL IDENTITIES"

PAVEL BARTOS, JOSFF KAUCKY, Bratislava

In the paper [1] we have described a certain method by means of which we can derive some combinatorial formulas. In this note we introduce another similar method.

T h e o r e m. Let n be a natural number, x an arbitrary coynplex number and d\, et2, ..., (iⁿ, O+fi the given distinct complex numbers, with the condition ak — a,k-n-\ for k > n + I. Then the, following relation holds

(x + a^t)(x + ai⁴1) ... (x + ai.n^i) (at - a^t A)(au^ri — (H-i) • • • (o-t^v,i i -- n+i)

Proof. (1) is an algebraic equation of degree n in v. But it lias (n -- 1) roots

(2) —-tfi, — a^{2 j} ..., a a , —«-,^Mi.

Therefore it is an identity.

In fact the factor (x + a^k), k ^ T, 2, ...,n, (n + 1) occurs in all members on the left side of this equation except in member with i k -[-- 1.^rrhus for x •— —a^k only the member

(—d k + <lk-\ l)(—(*-k + «>k\2) • • • (—(lk f dk,n)

(a^k 11 -- a^k)(a^{k] 2} — a^k) ... (a^k „ —- a^k) is different from zero.

E x a m p l e . Let a\ — i. In this case equation (1) gives

(x + l)(x + 2)...(x + n) (x + 2)(x + 3) ... (x -| /j i 1) ( ^ [ - ( „ „ ^ . . . ( - ^ ( - l ) 1.2

ЗҢa; + 4 ) . . . (aH- » + 1)^Ј- t I , (,r -|- 4)(

1.2 (n --- 1) I 1.2

+

^(a

Lt

³⁾

i? +

^{4 )}

•••

^(;r

+

^{и 4}

'

^{! )}

-

^Ј

'

^f

' •.

^(,ť

''"

^{4 ) (}

+

^{5 )}

••• <•"' f"" -

^{l )}

2 8 2

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(4)

(x + l)(x +2) x + n + 1 ( - l ) ( - 2 ) ' • •⁺ 1 (x+ l)(x + 2) . . . (x + re — 1)

\-(n - 1)] [— (n - 2)] . . . (—2)(—1) "

(x -j- k

£ - 0

In virtue of identity

ЂM- ^+l ){'**

^k ^1.

(«)

we have therefrom

x + n + 1\ lx + k\ __ íx + n + 1\ ln\ n + 1

n-k ) \ k ) ~ \ n+1 )\k)~~~~~~

(«)

2«-"',TíTi(í-[C::í>

_k--0

^+, >r

This is a generalisation of the well-known relation

(^)

**Уl-D*—**

¹

..- M____-

/ , n + k+ 1 ЏJ (2n + 1)

k 0

See [2J.

R e m a r k . Let us only remark that the identity (4) can be obtained in

n

another with the aid of Cauchy's identity *S (fy (^y \ = (* +^y)

k 0

w 1 ^ ^{; ; )} ^ ^J . ⁺ » ⁺ .-o,.. ^{+ 1} , ⁾ .2(-:»r)(- ^{T 1} )-

283

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KEFKRENCVS

J l | Bartoš I\, Ivane ký J., A (jenesis for comhinatorial identities. Mat.-ty/, časop. M {\\m\), 3 1 -4 0 .

[21 T i i n i n I \ , Probléme 51. Mat. lapok 7 (\\)W), 141.

K r c c i w d J u n e 2, HM>5:

in iv\ iscd 1'orm, J u l y 2(>. I9(>5.

ČSAV, Matonatickfi ústan Slorenskej akademie ricfl,

H)'atislara

284

Matematicko-fyzikálny časopis