a class of continued fractions

$: A class of continued fractions$
A CLASS OF CONTINUED FRACTIONS

BY H. M. SCHWARTZ

Introduction. The algebraic continued fractions

klZl t- - ..., k complex and 0,() I-i-- iwhose coefficients satisfy the condition

possess the following interesting property:The numerators and denominators of the approximans of (1) form respectively

two seuences that converge uniformly over any bounded region of the z-plane [3].A more general class than (1) is given by

k W k ]W kandccomplex, k0.(3) z v z cIn fact, except for simple changes of the variable and the fraction, (1) can beobtained from (3) by taking all c 0 ([4], 61). In this paper we study someconsequences of condition (2) for this general class of continued fractions.The general case of unrestricted c is considered briefly in 1. A generaliza-

tion of the above-mentioned property of (1) under (2) in this case is given in

THEOREM 1. Denote the n-th approximan of (3) by P(z)/Q(z). If we have(2), hen he two sequences

(4) P(z) Q(z) (n , 2, ...)H (z ) H (z ,)

converge each uniformly in every domain of the z-plane at a positive distance from{c}--the set of points c (i 1, 2, ).More precise results are obtained in 2 and 3 for a number of special cases.

Convergence properties of (3) are discussed in 4. In 5 we consider thespecial case corresponding to the condition

(s)

Received June 3, 1939; in revised form, November 10, 1939; presented to the AmericanMathematical Society, April 8, 1939, under a different title. The author wishes to expresshis gratitude to Professor J. A. Shohat for his encouragement and advice during the prep-aration of this paper, and for his valuable suggestions.

Numbers in brackets refer to the bibliography.By domain we mean a point set in the extended complex plane.

48

CLASS OF CONTINUED FRACTIONS 49

In this case (2) can be replaced by the weaker condition

(6)

Corresponding to Theorem 1 we have

If the coefficients of (3) satisfy conditions (5) and (6), then theTHEOREM 2.sequences

(7) P,(z) Q(z)

II (- c,) II c,)i,l il

(n 1,2, ...)

converge each uniformly over any bounded domain of the z-plane, the limit func-tions being, thus, entire functions.The second part of the paper applies to Bessel’s continued fraction results

obtained in the flint part. The related system of Lommel polynomials is shownto form an orthogonal set [5] for certain values of the parameter and the corre-sponding weight function is found.

I. A class of continued fractions

1. Proof of Theorem 1. Since

(8) P,(z) ]QQn-I.I(z) (n- 1, 2,... ),

it suffices to consider only {Q,(z)}.Let D be a given domain of the z-plane at the distance d (>0) from the set

(9)

By the Euler-Minding formulas

I 1,n.--Q,(z) II (z- c). 1 A-i1

ki-b(z c,) (z

1,n--2 ki-I ki+2- "i (Z C,)(Z Cl-bl) (Z C]-bl)(Z--" ’b2) -we have for all z in D

Q(z)

II (z

1, n--1 1, n--2

=-- Vn(z) <- 1 + k,+d----Y-I_

d+ II ki+2d _[_

the infinite product being convergent by assumption (2).rence relations

K, say,

Now by the recur-

(10) Q(z) (z c)Q_(z) - kQ_s(z) (n 2, 3,... )

Q,,,(z) (m 0, 1, ...) denotes, in Perron’s notation ([4], 5) the function arising fromQ,,(z) when/ and c are replaced by k+,, and c+ respectively.

5O H M. SCHWARTZ

we find

k, V,_(z) (n--2,3,...)V.(z) V,_,(z) + (z c,,)(z c,_,)

It follows that for all z in D

and so the series

V,(z) + [v,.(z) V(z)l +converges uniformly in D; that is, the sequence [V(z)} converges uni-formly in D.A necessary condition. When the set {ci} is unbounded, it is easy to see by

consulting Theorem 2 that condition (2) in Theorem 1 is not necessary forthe validity of Theorem 1. Indeed, by (5) it follows, as a consequence ofTheorem 2, that the sequences

P,(z) 1 Q,(z) Q,(z) 1

H (-c,) II 1--z II (z-c) H (-c,) II 1--

z

converge uniformly in every bounded domain of the z-plane at a positive dis-tance from {c}, while (6) shows that the set {k} could even be unbounded.

Consider, then, the case of bounded {ci}. By (9) we have for all z at a suffi-ciently great distance from the origin, the following expansion in ascendingpowers of z-1"

1, --1 I! CiV(z) l + . + + I I Cil/ -V / )By Weierstrass’ double series theorem, it follows, therefore, that when {c} isbounded, then for the conclusion of Theorem 1 to hold true, it is necessary thatk converge.

Note. In the important special case

(11) kl > 0, k < 0 (i- 2, 3,... ); c real (i-- 1, 2,...

which is related to the moments problem of Stieltjes and Hamburger ([4], 72)it is thus seen that condition (2) in Theorem 1 in case of bounded {c} is bothnecessary and sufficient.

2. When in addition to (2) it is also assumed that

(12) lim c exists and c, say,

it is possible to obtain a more precise result than that given in Theorem 1.We make use of the following


LEMMA. If the sequence of analytic functions f,(z) (n 1, 2, ) satisfies thefollowing two conditions:

(a) f,(z) (n 1, 2, ) is everywhere regular, except possibly at a given point c;(b) there exists a sequence of closed contours C,, (n 1, 2, ) enclosing the

point c, with

lim max z-c[ =0

and such that on each C, the sequence {/(z)} converges uniformly;then

(c) sequence {f,(z)l converges uniformly in every domain at a positive distancefrom the point c, the limit function being an entire function in (z c)-1.

The lemma is an immediate consequence of a well-known theorem of Weier-strass, which states that a sequence of functions regular analytic inside and on agiven closed contour, and converging uniformly on the contour, converges uni-formly in any closed domain in the interior of the domain bounded by thecontour. We have only to subject the z-plane to the linear transformation

1

and to observe that the functions

(1)are, by condition (a) of the lemma, everywhere regular in the finite zr-plane,and that, according to condition (b) of the lemma, any finite domain of thezr-plane can be enclosed in a closed contour (namely, the map in the z’-plane ofone of the contours Cn given in (b)) on which the sequence converges uniformly.By the above theorem, it follows that {,(z’)} converges uniformly in anybounded domain and its limit function, therefore, is an entire function. Trans-lating this result back into the z-plane, we obtain the conclusion (c) of thelemma.

Consider now the sequences (4) of Theorem 1. It will be sufficient to con-sider one of them, say the second. By condition (12) and Theorem i (condition(2) being assumed), it can be readily concluded that we can find a sequence ofcontours C (each Cn being at some positive distance from the set {c}) whichsatisfy the conditions given in (b), with

f(z)Q,(z)

II (z c,)

In our application, it is sufficient to take for C. a circle with c for center and radiuswhich --, 0 as n

52 H. M. SCHWARTZ

Suppose now that the set of integers {p} (i 1, 2, ) is such that

(13) c cconverges for every z c

-1 Z C

(it being always possible to determine such sets of integers, since by (12)lim (c c) 0), so that the infinite product

[: (:’::) +... -(: :)’1I 1 c-- exp --c+l 1-, z- -5 c +p

converges uniformly on each of the above contours C,. By the identity

(14) Q,(z) er(’:") Q,,(z)

I (z c,)(z c)’* er(,;, II 1

c,-

i,l i-’l Z

where

(15) T(z; n),-1 L------c + + + p

it follows that condition (b) of the lemma holds also for the sequence(z;n)e(16)

(z c)Q,(z) (n 1, 2, ...).

But this sequence satisfies also assumption (a) of the lemma, the functions Q,,(z)being polynomials in z. Hence, for sequence (16) conclusion (c) of the lemmais valid and we obtain

THEOREm 3. If the coecients of the continued fraction (3) satisfy conditions(2) and (12), then sequence (16) and sequence

P,(z) (n 1, 2, ...),(z c)"

where T(z; n) are given by (15), converge each uniformly in every domain of thez-plane at a positive distance from the point c, the limit functions being entire func-tions in (z c)-1.COROLLARY. If we can take p p (i 1, 2, ...) in (13), and if we have

further

(17) (c c) converges for k 1, 2, ..., p,

then the two sequences

converge uniformly in every domain at a positive distance from the point c.


In fact, we have only to note that by virtue of assumption (17), the followinglimit exists"

tim T(z;n) 1

this limit being, clearly, uniform with respect to z in every domain at a positivedistance from the point c.

If instead of (12) we suppose that

(19) lim1

then, by examining the proof of Theorem 3, it is easy to see that the argumentof that proof can be retained.itself rather than the lemma.consider the following identity

Q.(z)

II c,)

where

(20)

In this case, we have to use Weierstrass’ theoremMoreover, in place of identity (14) we have to

eR(.;.) Q.(z) 1

(--c,) eR(’;’) II 1- z

R(z; n) [z 1() 1,_, ;+ + + j,

the integers p being determined so that

Z’p/’ +1

(21) converges for every finite z.

THEOREM 4. If the coecients of the continued fraction (3) satisfy conditions(2) and (19), then each of the two sequences

where R(z; n) are given by (20) and (21), converges uniformly in every boundeddomain of the z-plane, the limit functions being thus entire functions.

If we have further p p (i 1, 2, ) and

1- converges for k 1, 2, ..., p,i,-1 Ci

then the above convergence behavior belongs already to the sequences (7).

3. The condition

(22)

54 H. M. SCHWARTZ

is included as the special case p 0 in the corollary to Theorem 3. This case,however, can be proved directly by very simple means.

First, we restate it, for the sake of convenience, as

THEOnE 5. If the coecients of (3) satisfy conditions (2) and (22), then eachof the sequences (18) converges uniformly in every domain of the z-plane at a positivedistance from the point c.

Q,(z) Q*(z) (n 1 2,... ), we find byWriting for the sake of brevity(z c)

the recurrence formulas (10)

z c/ (z c)Q,,_.(z) (n 2, 3, ...).

\

Therefore, for all z in a given domain D at the positive distance d from thepoint c,

Q,-,(z) <-_ A.d

(n 2,3, ...),

where

A {IQ*(z) I}.inD

n--1,2,

Moreover, by formulas (9), we have in D

(23)

*Q,,(z)=II 1sl

1 1,n--1 1,n

si,i-bl

1,n--21ki+: k,i+. II

Consequently, by assumptions (2) and (22), we have for all z in D and for all n

IQ*(z)l<II 1T Ic’-cl l-t-- d T d d T...

(24) H 1T ]c’-c 1+

H 1+ lc’-c 1+

It follows that A is a finite number, and the proof of Theorem 5 may be com-pleted exactly as the proof of Theorem 1.The corollary to Theorem 3 shows that assumption (22) is not necessary for

the validity of Theorem 5. However, the weaker assumption

(25) (c- c) convergesi1


is indeed necessary. In fact, we have

THEOREM 6. If one of the sequences (18), say Q*(z), converges uniformly insome region D of the z-plane at a positive distance from the point c, then we musthave (25).

This follows by the application of Weierstrass’ double series theorem to theexpansion (cf. (23)), 1 1 (c- c,)(c ci) -{- k -{-Q,(z) 1 T z- c

(c c,) + (z c) ,<

We note further that the existence of the limit for n -- of the coefficient1,n

(26) (c c,)(c c) T k,

of (z c)-2 in the above expansion wil! imply the condition

(27) k converges

if we assume (22), sinceI,

5:1( c)(c c.)l

_(5: i , !).

It follows, therefore, that

when the coecients of fraction (3) are such that the convergence of the series, (c c) implies their absolute convergence (as might happen, for instance,when we have the conditions (11)), then the conditions (2) and (22) of Theorem 5are necessary as well as sucient for the validity of that theorem.

SinceI,

Z: ( ,) [Z: (c c)]- 2 ( ,)(c c,),

it s seen by Theorem 6 and by the form of the coefficient (26) hat

if we assume (2) or only (27), hen he assumption in Theorem 6 implies inaddition to (25) also

(c c,) converges.

On the order of the limit functions of Theorem 5. It will be again sufficient toconfine our attention to one of the sequences (18), say to {Q,*(z)}. Let

(28) lim Q*(z) Q(z)

By (24), we find, r being an arbitrary positive number,

max IQ(z) i<_-II l+lC-C i+

5 H. M. SCHWARTZ

Hence, if we denote by pl and p. the exponents of convergence of the two series

E (c,- c), Erespectively (the exponent of convergence of a series a being defined as

the greatest lower bound, if it exists, of the set of numbers {/} for which

la [k converges), it follows by a familiar reasoning in the theory of entire

functions (namely, that concerning the order of a canonical product) that givenan arbitrarily assigned positive number e, we can find a positive number r suchthat for r =< r we have

where

max Q(z) < exp I 1

p* max {pl, 2p.}.

By the definition of the order of an entire function, we have thus

THEOREM 7. The order p of either the limit function (28) or the limit functionP (z)P(z) .-lim (z c)

of Theorem 5 satisfies the inequality

p __< max {pl, 2p.},

where pl and p. are defined above.

therefore, we have

In the special case

c c (, , 2,...

(29) p =< 2p..

In view of (2) and (22), we have in all cases

p<--2.

In the particular case (11), which, as was mentioned before, is related to themoments problem of Stieltjes-Hamburger, it is easy to obtain additional infor-mation concerning the limit functions P(z) and Q(z), by taking in account theknown fact that the zeros of P,(z) and Q,(z) are in that case all real ([4], 69;[5]), and by referring to the following theorem of Laguerre:

The limit of a sequence of polynomials having only real zeros, which convergesuniformly in every bounded domain of the z-plane, is an entire function of genus 1at most, except for a possible factor of the form eaz, where a is real.

A similar result for fractions (1) under (2) is given in [3].Oeuvres, vol. 1, pp. 174-177.


Since P(z) (z c) and Q(z) (z c) are (in case (11)) polynomials in(z c)-1 having only real zeros, we obtain immediately

THEOREM 8. When the coeiicients of the continued fraction (3), in addition tosatisfying the assumptions given in Theorem 5, also satisfy (11), then the limitfunctions P(z) and Q(z) of Theorem 5 are each at most of genus 1 in (z c)-except for a possible factor of the form

expla/z 1 till, areal.

The case c c. The zeros of Qn(z) are, in this case, symmetric about thepoint c, and it is seen that

Q,(z) I [(z- c)- z,..],il

(z c) [(z c)

so that

(c :i: z, are the zeros of Q,(z)),

Q,(z)(z ._ (z =

It follows that the limit function Q(z) is even, considered as a function of(z c). In the special case (11), it is then seen by Theorem 8 that Q(z) is ofgenus 0 in (z c)-2 except for a possible factor of the form exp [a(z c)-2],a real. By utilizing also the separability properties of the zeros of Q,(z) under(11) [5], it can be shown independently of Laguerre’s theorem that in this caseQ(z) is exactly of genus 0 in (z c)- (i.e., a 0).

If we do not assume (11), then the above result can still be retained providedwe suppose that p. of Theorem 7 is -< 1/2. Indeed, this latter assumption impliesby. (29) that the order of the limit function Q(z) is -<_ 1. Hence, since the limitfunction is even in (z c), we have

THEOREM 9. The limitfunction Q(z) of Theorem 7 will be of genus 0 in (z c)-2

when c c and when p. given in Theorem 7 is <= 1/2.

4. By Theorem 3, it is immediately apparent that the continued fraction (3),with the coefficients stisfying (2) and (12), converges at every point of thez-plane with the possible exception of the point c und of the zeros of the limitfunction of sequence (16); and that it converges uniformly in every domuin ofthe z-plane at positive arbitrarily small distance rom the point c and fromthose zeros. By Hurwitz’s theorem concerning the zeros of the limiting func-tion of a convergent sequence of analytic functions [2] it is readily seen that the

By means of the recurrence formulas (10) (and Q 1, Q z o) we find

Q,,(c -t- z) (-1)n@.(c z) (n 0, 1, 2, ...).

Cf. [6], p. 529, where a similar result is obtained for (1) under (2).

58 H. M. SCHWARTZ

zeros in question are given by the limit points of the set of zeros of the poly-nomials Q(z) (n 1, 2,... ), points which are zeros for an infinity of Q,(z)being considered as limit points.

Convergence of (3) under the assumption of 3. We make use of the followingtwo theorems:

THEOREM 10.zeros,l

The limit functions P(z) and Q(z) of Theorem 5 have no common

THEOREM 11. Given an arbitrarily assigned positive e, we can find a numbern, such that for all n >= n the zeros of the polynomials Q,(z) of Theorem 5, whichlie outside the circle z c e do not coincide with any zeros of the limit func-tion Q(z).Theorem 10 can be proved in a manner analogous to Perron’s proof of Mail-

let’s theorem ([4], p. 346, footnote). In fact, writing (the existence of thelimits is obviously assured by Theorem 5)

Q,(z)lim

(z c)"qm(z) (m O, 1,... ;z c),

we can show that

(30) q(z) z c+q+(z + +z c (z v)2qm+(z) (m O, 1, ...),

(31) lim q(z) 1, z c,m--

and then apply Perron’s reasoning.To prove Theorem 11, we consider the following formula ([4], 5)

Q,+,(z) Q(z)Q,,,(z) + k,+lQ-i(z)Q,-l,,+l(z).

Dividing on both sides by (z c)n+ and letting m -- , we get

Q(z) * + *Q,,_(z)q+l(z)Q(z)q(z) + (z c)Let, now, e (>0) be arbitrarily assigned. By (31) we can find a number n,such that for n >= n,, q,+(z) has no zeros outside the circle z c e. Itfollows, that for the same n (i.e., for n _>_ n,) if i" is an arbitrary zero of Q(z)such that I c > e, we cannot also have Q() 0. For, if the latterequation were true, then it would follow that also Q-(i’) 0, and this isimpossible in view of the relations ([4], 6)

P,(z)Q,_(z) P,_l(z)Q,(z) (-1)-k. /c

and our assumption k 0 (i 1, 2, ).The limit function of sequence (16) is not 0, for

lira Q.(z)-- 1.-1- (z c)

10 An almlogous theorem for (1) under (2) was first proved by Maillet [3].


By Theorem 10, it follows that the zeros of Q(z) are points of proper divergenceof the continued fraction (that is, at those points, the reciprocal of the continuedfraction converges to zero). Furthermore, by Theorem 11, it follows that thelimit points of the set of zeros of the Q,,(z) (n 1, 2, can be taken in thegeometric sense, since no points of the z-plane distinct from the point c can be azero for an infinity of Q,(z). We have, therefore,

THEOREM 12. Under the assumptions of Theorem 5, the continued fraction (3)converges at every point of the z-plane with the exception of the limit points of theset of zeros of the polynomials Q,(z) (n 1, 2, and with the possible exceptionof the point c. At the aforementioned limit points, the continued fraction divergesproperly. In every closed domain excluding those limit points and the point c, thecontinued fraction converges uniformly.

5. Proof of Theorem 2.

V(z) Q,(z) II (- c,),

By the recurrence formulas (10), we have, putting

Hence

V(z) 1- c,-xc,,

V.(z) =< i-C:. +On the other hand, it is seen, as in the case of the derivation of (24), that

Recalling assumptions (5) and (6), we see that the theorem follows just as inthe proof of Theorem 5.A partial converse to Theorem 2 for a special case is given in the following

remark:

If the coefficients of (3) are real, and if

then the assumption

(32)

n+l >0 (n--- 1, 2, ...),Cn Cn+l

lim Q.(0)"-’ I (-c,)

exists

implies (6). Moreover, if we have

k, > O, c, of same sign (n 1,2, ),

0 H. M. SCHWARTZ

then (32) and the additional assumption

limQ(0)

exists

imply both (5) and (6).

To see the truth of this remark we have only to observe that by (9)

Q,,(z) II 1-z ""-’ k,+l

II(-c,) -1+ II 1-

z

i--1

and therefore

Q.(0)

II (-c,)

I,n--2 I,-- Z ]-[-I ]-{-I Hi<" CiC/+l C/+lCi+$

Q(O)k=4 Ck CiC/+l ki,i-t-1 Ck

By examining the proofs of the theorems given in 3 and that of Theorem 12,it is readily seen that we can prove by the same methods, analogous results forthe continued fraction of Theorem 2.

II. Bessel’s continued fraction and Lommel polynomials

6. An interesting example of a continued fraction belonging to the class dis-cussed in 3 is provided by Bessel’s continued fraction ([7], 5.6)"

(33) 1/2 [_ 1/4( T 1)[_ 1/4( -[- 1)( -t- 2)[

z 0, complex and 0, -1, -2,

which is "equivalent" ([41, 42) to"

1 1 1(34)12/t 120, / 1)/t -’z

The denominator of the n-th approximant of (34) is Lommel’s polynomialn

11 This designation is slightly inaccurate, Rn,,(t) being a polynomial in 1/t add not in t.


R.v(t) ([7], 9.6).12 An important result, discovered by Hurwitz [2] and appliedby him to the investigation of the zeros of Bessel’s function Jv(z), is

(1/2t)"+(35) ,-lim I’(n + u -- 1) R,.,+(t) J(t)

for non-integral values of u, the convergence being uniform in any boundeddomain of the t-plane.As an application of some of. the topics discussed in part I, we shall give a

simple proof of (35) with the u restricted only as in (33)" u 0, -1, -2,Other known properties of J,(z) will also follow from the development that willnow be outlined.The explicit expression for J,(z) will be needed only at one step in our dis-

cussion where we have to identify it with a derived function. Otherwise, ourinformation is obtained directly from the functional equation ([7], 3.2)

(36) J,-x(z) A- J+l(z) 2 J(z).z

By (36) it is easy to verify the known fact13 that the continued fraction (33)represents the function J(z-)/J,_x(z-) F,(z). Indeed, it is readily de-ducible from (36) that the function14 F,(z) is "associated" ([4], 61 and 67)with the continued frac.tion (33). But this fraction represents, by Theorem 12,a meromorphic function in z-, and it can be easily concluded, taking in con-sideration the uniqueness of "associatedness" ([4], 61), that this meromorphicfunction must be identical with F(z).5

Consider now fraction. (33) and denote its n-th approximant byP,(z; ,)/Q(z; ,). By Theorem 5, the following limits exist"

(37) lim P,(z; ) pp_x(z), say; liraQ,,(z; ) q_x(z), say.

n-* Z n

Again, by (8), it is seen that

1P.(z; ,) Q._,(z; + ).

1 This is not the usual definition ([7], 9.6), but it follows readily from the followingproperties of Lommel polynomials ([7], 9.63)

2( + n 1)R,.,(t) R,_.,(t) R.,.,(t) (n 2, 3, ...),

Ro,,(t) 1, Rx,,(t) ----.

[4], p. 299.That is, its formal expansion in powers of z-1.Cf. [4], p. 353 and Theorem 20 on p. 342.

Hence1

and we have (by the above)q,(z) (z 0).F,(z)

2,,z q,_(z)

If we put q,,(z)/J,,(z-) X,(z), the above equation becomes

X,(z) 2zx,_,(z),

a derence equation in v having the solution

1X,(z) (2z)’r(v + 1), ,+(z) ,(z).

Thus:1(38) g (z)

2r( T 1)zq(z) (z 0).

It will now be shown that (z) 1, by identifying the coefficients of thepower series expansion of q(z) with those of 2F( T 1)z’J(z-). In fact,consider (30) with m 0. It is readily seen that in our case it can be written

1q,_,(z) q(z)4( + 1)z

q+,(z).

Let the expansion of q(z) be (the form following by 3)

1 + q,)l 1+q<’)+Substituting in the above equation, we obtain the system of difference equations

q) _(,_) 1 q) (m 1,2, ...),4r(r + 1)

and these lead very easily to our desired result.

Since q’) 1, and since (of. (9))Q(z; r)

1 + 1 1,n--1 1,n--2 1

1 1kffi-- k= (iffi2,3, .’.),2’ 4( + i 1)( + i 2)

taking m 1 gives

q,) 1 1 1

4 ( + k)( + + 1) 4’

the periodic additive function vanishing by (37). Proceeding in this manner, we find (bymathematical induction)

(-1) 1

4m ( + 1) ( +m- 1)"


We have now by (38)1J(z)

2F(v - 1)z"q(z).

This relationship permits the application of the results obtained in 3 to thefunction J(z). Thus, Hurwitz’s result (35) follows immediately from (37),since

2n(-{- 1)... (-{-n- 1)Qn(z; )

by the definition of R. and Qn ([4], 42), while the well-known fact that z-J(z)is of genus 0 in z follows from Theorem 9.

7. For positive values of the parameter , the coefficients of the continuedfraction (33) are seen to satisfy conditions (11) (with c 0) and the fractionis therefore related to a moments problem ([4], 72; [6]; [1]; [5]). That meansthat if the power series "associated" with (33) is

a0 _. al a.- ++"’,then the infinite system of integral equations

(40) x de(x) a (n 0, 1, ...)

has at least one solution (x) which is a bounded monotonically increasing func-tion of the real variable x having an infinity of jump-points, the integrationbeing taken in the Stielties sense. It follows further that the sequence of poly-nomials Q,(z; ,) =- Q,(z) considered in 6 forms for every > 0 an orthogonalset with respect to this moments problem [5]; that is,

I Q,(x)Q,(x)de(x) 0 (n m; n, m 0, 1,...),

where (x) is a solution of (40). Hence by (39), we have

f 1-)(41) R,, R,. d4(x) 0 (n m; n, m O, 1, ;, 0).

This result is a new property of the system of Lommel polynomials.By (41), we can obtain very simply the reality and separability properties of

the zeros of Lommel’s polynomials for > 0 [5] and thence the reality of thezeros of J(z) for real and > -1. In fact, we can apply the extensive theoryof orthogonal polynomials [5] to obtain further properties of R.. However,for a full utilization of the relations (41), it is necessary to solve the momentsproblem (40). This can be accomplished very easily in the present case. Thefact, pointed out in 6, that the fraction (33) represents a meromorphic functionin z-1 tells us immediately ([4], Chapter 9; [5]) that the "interval of orthogo-

64 H. M. SCHWARTZ

nality" [5] is finite and the moments problem is "determined" ([4], 72; [5]);that is, there exists essentially only one solution. It also follows readily by"Stieltjes’ inversion formula" ([6]; [4], 66)7 that the solution (x) is a step-function with jump-points at the poles of the meromorphic functions in ques-tion,is namely, J(z-)/J_(z-) (cf. 6). Furthermore, since all ci 0 it follows[5] that the solution is "symmetric"; that is, we huve with the normalization(x) 0

(42) (x) --(--x).

To solve our moments problem completely it is still necessary to determinethe jumps of O(x). Denote the positive zeros of J(z-) by 3) j (k1, 2,... ;j+ < j) and the corresponding jumps of (x) by h. By (42),it follows that the jump-points of (x) are j ( 1, 2, and the intervalof orthogonality is (-j, J0. Now, by Markoff’s theorem ([4], 68) the con-tinued fraction is equal (at its convergence points) to the Stieltjes integral

’ d(x)

Therefore, by 6 and the form of (x),

J ; d(x) h

the series being absolutely convergent for all z 0, j, j, It followsthat h is equal to the residue of the function J(z-)/J_(z-) at the point jso that, remembering the definition of j,

1h lim (z-j) J

-7;_() + 7._() 7,(),

h j (k 1, 2,... ).

The orthogonality relations (41) have thus the following form"

jRn., ., =0 (mn;m,n=O,,...).

17 Hilfssatz 5. Cf. also A. Wintner, Spektraltheorie der Unendlichen Matrizen, 43.is Cf. [4], p. 403, and A. Wintner, loc. tit., 45.


BIBLIOGRAPHY1. H. HAMBURGER, Ueber eine Erweiterung des Stieltjesschen Momentenproblems, Math.

Annalen, vol. 81 (1920), pp. 234-319; vol. 82(1921), pp. 120-164, 168-187.2. A. HURWITZ, Ueber die Nullstellen der Bessel’schen Function, Math. Annalen, vol.

33(1889), pp. 246-266.3. E. MAILLET, Sur les fractions continues algbriques, Journal de l’]cole Polytechnique,

(2), vol. 12(1908), pp. 41-62.4. O. PERRON, Die Lehre yon den Kettenbrichen, 2d ed., 1929.5. J. SHOHAT, ThOorie GnOrale des Polynomes Orthogonaux de Tchebichef, Mmorial des

Sciences Mathmatiques, Fasc. 66, 1934.6. T. J. STIELTJES, Recherches sur les Fractions Continues, Oeuvres, vol. 2, pp. 402-566.7. G. N. WATSON, Theory of Bessel Functions, 1922.

UNIVERSITY OF PENNSYLVANIA,

a class of continued fractions

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