Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1991. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 20)

Kristyna Grytczuk: Functional récurrences and differential équations

- 53 PROOF OF THEOREM B. Let us denote LCy> = P v

c n

5 + P y

t

n"

1

J + ...+ p y J O l

J Tt

J then for y = s„ we have 1 o, 1 1 = L(

s0. i'

ui) = PoKt*

UÍj

tn5

+ ••• - PnKl'

Ut]It is easy to see that is *uM' = s

' *u* + \*s •u*~

1

,u' = u* is' L o. l lJ o, i t o, 1 1 1 l [ o, i * u

4J Putting in the last equality s' + x * — = s O, 1 ^ YL

± 1. 1 we get is = s u* . L o. 1 1

J 1.1 1 Similarly we get is -uM

Ck

5= k -uM'-s, -U* [ O, 1 1 J ^ k - 1 , 1 1 J k . 1 1 for k—2

, 3, . . . , n

. Thus we get Lf

s« 's f 'u*+...+P *s *u*= L 1

. O 1J O n.l 1 1 n-1,1 1 n O, 1 1 = fp *s +P *s + ...+P *s . L O n.l 1 n - 1 . 1 n O, 1J 1 We remark that from (d) (6) det S = 0. follows. On the other hand from <d> we have C

7) Det S = s *D -s . *D +...+s

n C-l> D, n, 1 1, O n-1,1 1,1 O, 1 1, n • 1 Since P._

t = <-l)

j D

4 for j—

0,1,...,n+1 thus by C6) and C75 it follows