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

Kristyna Grytczuk: Functional récurrences and differential équations

- 32 u * i = s' + X s , 1 o ^ o u ' u * s2 = + ^ si u V S = + ^ lo r ' t = t' + X t — . 2 1 IV Theorem A has been used to investigation of some functional recurrences connected with number theoretic polynomials, in particular Pell, Lucas and Fibonacci polynomials. In the present paper we give some generalization of Theorem A. Namely we prove the following theorem: THEOREM B. Let u, be functions of x for k=l

,2,...,n o, k ' k and let X be a constant satisfying the following conditions: . - C

Cn

3CJ), where J=Cx

f

.x

2>cR for k=l,2, • k ** 0 on J for k=l,2,...,n and X « R

+. Then the functions Ca) Cb) so. k>

u> u, n; • u* o, 1 1 V O. 7- 7. Cl) y . = s_ . • u'; , y

7

= s are the solutions of the differential equation C2) where C3) P._

t = C-l)

j *D

1 .; j=l,2,.

..,n+l y - s -u

X ' r> O, n n P y

Cn

i+P y

C

n"

1

5 and D

t t denotes the minor of the matrix S which we get by deleting the first row and j-kolumn of the matrix S, where C4) n, 1 n. 1 n, 2 n- 1 , 1 n - 1 . 1 1. 2 and C3) for 1,k=l

,2,...,n o. 1 o, 1 o. 2 s s n

, n n - i , r> si

, k ~ si-l.k + X*

si-1. k ' U. O. N