MA THEMA TICS: SHOBA T A ND SHERMAN
VOL. 18, 1932
283
ON THE NUMERATORS OF THE CONTINUED FRACTION - x2 1
Ix
-
C2
...
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MA THEMA TICS: SHOBA T A ND SHERMAN
VOL. 18, 1932
283
ON THE NUMERATORS OF THE CONTINUED FRACTION - x2 1
Ix
-
C2
-...
BY J. SHOHAT AND J. SHERMAN DEPARTMENT OF MATHEMATICS, UNIVERSITY OF PENNSYLVANIA
Communicated January 21, 1982
1. It is known that the infinite continued fraction K
=
Ix _'
-
|x
(i, ci real;
-
X.>O°)
(1)
is. associated" with one and only one "positive definite" power seriesl2 p (X 3
E i+i Wth
II ai+j || i"
_ > O (n ) )
(2,
and with at least one Stieltjes integral of the form F(x)
df (y)
(3)
where t(y) is bounded and non-decreasing in (- co, co), with infinitely many points of increase [0(- co) = O!] and is a solution of the "[a] c' moment problem," i.e., J
x#du(x)
=
a,
(nih moment) (n = 0, 1, 2, ...).*
(4)
The interval (- c, o) in (4) may be "reducible," i.e., it may reduce to a sub-interval (a, b), if 4,i(x) = ,6(a) for x < a and 14(x) = A,(b) for x > b. (n = 0, 1, 2, ...) the successive convergents to (1). Denote by * From the very definition of "association" of continued fractions of type (1) with power series (2) and integrals (3), namely, ai UM(x) = a'~ ~ f ±..., (5) E i=O x+ - i+1 (x) x2n+1 ± x2n+2 co
we derive the fundamental orthogonality relations characterizing the
4,(x):
co
J m(x) O(X)
d4(x)
=
O (m n; m, n
= 0,
1, 2,...).
(6)
The object of this Note is to state some general properties of the numera* We may have, in particular, d*(x) = P(x)dx, P(x) 2 0.
MA THEMA TICS: SHOBA T AND SHERMAN
240u
PROC.. A. S.
tors 2, (x), which so far have been studied in very special cases only. The most remarkable property is expressed in THEOREM I. The numerators n(x), (n = 1, 2, ...) of the successive convergents to the continued fraction (1) form, like the denominators, an orthogonal system of polynomials, i.e., there exists a function y6I'(x) of the same nature as A,t(x) in (4), such that co f £nm(x) (n(x) d4ki(x) = 0 (m # n; m, n = O, 1, 2, ...), the interval (- , co ) perhaps being "reducible" for ,61(x). In fact, we have the following easily provable
A" A': LEMMA. Denote by A- and A", respectively, the successive convergents to n
_ail
K' 3=|bil + a2 +
l1 Then A,"+,
b2+
n
_2 ±-as +
K.. a + a
.
= aiB' (n = 0, 1, 2, ...).
(a,, bi arbitrary, ai 0 ).
.
lb2 lb3
It follows that the polynomials
Un(x) are, disregarding constant factors, the denominators of the successive convergents to
|X-C2 |X-C3
K
(7)
X
which, being of the same type as (1), gives rise to relations simnilar to (5, 6), and this proves our statement. We may write, without loss of generality, co
-
F(x)
1
2
r-IYC.. x y EoX+1
(aj = Jixid+(x); ao = 1) (8) _co
F, (x)
=
-,
i-0x+l|xc|-C
(i= jixid&l(x); #o 1), (9) CO
and we thus get (as in (6)): co
m(x) a. (x) dFl (x)
=
an
O (m 0 n; m, n
=
O, 1, 2,...) ... ;
We also get, introducing the "normalizing factors"3 4 = (1-X2X3 ... cn X* +i) /2; (laX' -2) / =
n>0
(10)
V/2a.+1: (U)
VOL. -4, 1932
MA THE MA TICS: SHOBA T AND SHERMAN
0f (m(x) (pn(x) d{t(x) =
285
Xm(x) X.(x) d41(x) = em,, (m, n = 0, 1, 2,...) So"(X) a,4,(x), X"(x) = a"o,(x). (12)
2. Hereafter t,'(x) denotes generally a solution of the [a,-]- moment problem related, in view of (5), to K in (1). We need some properties of the zeros of t,(x) and Qn(x) which we denote, respectively, by xi, x, (i = 1, 2, ..., n; j = 1, 2, ..., n - 1).1,2,4,5 Xi' X' X' ()Xl