a matrix representation of ascending and descending continued

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a matrix representation of ascending and descending continued
Algebra und Zahlentheorie
A MATRIX REPRESENTATION OF ASCENDING
AND DESCENDING CONTINUED FRACTIONS
By L. M. MILNE-THOMSON, Greenwich
The elements of a matrix, which is equal to the continued product of suitably
chosen matrices, are shown to have the properties of the convergents of a continued
fraction. The method leads at once to a generalised definition of continued fractions
in many dimensions. By turning through two right angles the plane on which an
ordinary (two dimensional) continued fraction is written, an ascending continued
fraction composed of the same elements is obtained. This again can be represented
by a product of matrices and generalised to many dimensions. Some properties and
applications of continued fractions of both types are considered. A paper dealing
with this subject will be published in the Proceedings of the Edinburgh Mathematical
Society.
LE SERIE RICORRENTI ASSOCIATE DEL 2» ORDINE
(Generalizzazione delle Un e Vn di Lucas)
Di GIACOMO CANDIDO, Brindisi
i- Defin. Due serie del 2° ordine
WB, wlt
entrambe riferite
....,
WH,
....(W)
¥0 , Wlt ....,
Wn ,
....(¥)
alla stessa equazione
g—PÌ^
caratteristica
Ç = o .... [ i ] ,
coi rispettivi valori iniziali W0, Wx ; ¥0, Wx le diremo associate se una coppia
qualunque ( Wz-, ¥z- ) di termini corrispondenti verifica la identità
Vnp,q)-àWi(p,q)
2
con A =p
= kq<, ....[E]
— 4 q9 e k costante.
E s : i) Le Vn ed Un di Lucas. 2) Le soluzioni della equazione x2 — Ay1 = kq*
2. L'integrale generale della [W) è dato da (Eulero, Le Longchamps, d'Ocagne):
Wn (p, ç)=W±l7n
— p W0 £/„_!
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