Polynomial sequences with prescribed power sums of zeros

Journal of Computational North-Holland

and Applied

Mathematics

317

49 (1993) 317-327

CAM 1438

Polynomial sequences with prescribed sums of zeros

power

H. Van Rossum Utrecht, Netherlands Received

8 November

1991

Abstract Van Rossum, H., Polynomial sequences with prescribed Applied Mathematics 49 (1993) 317-327. Let zl, z~,..., z, denote the zeros of the complex called the kth power sum of zeros (psz) of V,(z). problem. Construct a polynomial 1/,(z) given its required V,(-- z)/I/,(z) to be irreducible. The -3+

of Computational

and

polynomial V,(z). Put a, = zt + 2,” + . . . + z,k, k E N; a, is Laguerre (in 1880 and 1887) posed and solved the following odd psz cl, F~,. ..,uZnpl. To ensure a unique solution he solution is the (n, n)-PadC denominator to exp(2(cr,z-’ +

-Zn+l)}.

. . .

(+3z

power sums of zeros, Journal

+(+2n-1z

The special case where u1 = - f, u3 = u5 = . . = uXn _ 1 = 0 yields the ordinary Bessel polynomial of degree II. With exception of their orthogonality, Laguerre gave the first treatment of the orthogonal polynomial system (OPS), called Bessel polynomials, much earlier than Bochner (1929) and Burchnall and Chaundy (1931) did. In the present paper we generate OPS by prescribing the odd psz: an infinite sequence ul, u3,. . , uZn_ 1,. . . . Results on the differential equations and zeros are given. Finally, a generalization of ordinary Bessel polynomials is given, such that the defining property of being an OPS with prescribed psz is retained. Keywords: Power sums of zeros; Padt

table; orthogonal

polynomial

systems;

Bessel polynomials

1. Introduction

Let the complex polynomial Vn/,(z)have zeros zl, z2,. . . , z,. Put a, = zt + zf + . - * +z,k; ak, of zeros (psz> of V,(z). Laguerre [6, pp. 119-1221 sought to determine a polynomial V,(z), with deg V,(z) Q n, by prescribing the psz ul, c3,. . . , u~~_~. He only considered odd psz, as we will do also in the present paper. The solution to Laguerre’s problem in general is not unique. If deg V,(z) = n, presents an infinite set of solutions. Laguerre [6, pp. 119-1221 required that V,(-z>/V,(z> be irreducible. His solution can be phrased as follows. If the psz (TV,u3,. . . , a,,_ 1 (not all equal to zero> are given, then the polynomial K’,(z), k E N, is called the kth power sum

Correspondence

to: Prof. H. Van Rossum,

0377-0427/93/$06.00 0 1993 - Elsevier SSDZ 0377-0427(93)E0065-T

Winklerlaan Science

92, 3571 KL Utrecht,

Publishers

Netherlands.

B.V. All rights reserved

H. Van Rossum / Polynomial sequences with prescribed zeros

318

deg V,(z) G II, with these prescribed psz is the denominator of the (n, n&Pad6 approximant to f(z) given by 1 f(2) = exp 2 aIz-l+ $,zp3 + *** + 2n_102~_Iz-2”+1 (1.1)

II,

ii

i.e., we have f(z)

u,(z)

=

K(z)

+

2n-1

w-

)a

The nonnormality of the (main diagonal) of the Pad& table for f(z) may cause deg V,(z) < n. From the theory of Pad6 tables it follows that in (1.2) we have U,(z) = V,(-z). We generalize the prescription of psz for one polynomial to that for an infinite sequence of polynomials. sequence (V,(z)>“, = 1 is said to possess prescribed psz ul, if (1) for every m E N holds (TV,g3,. . . , uZm_ 1 are the psz of V,(z); 2m-‘/(2m - 1) < CCon some disc 1z I< 1, 1> 0; (2) C;=l54Z (3) the main diagonal of the Pad6 table for the function m 1 exp 2 C ~~~~~~~~~~ z+ 2m-1 I i m=l is normal.

Definition

1.1. The polynomial

C33,...,U2m-l,...

Theorem 1.2. The polynomials V,(z), m = 1, 2,. . . , with prescribed psz crl, u,, . . . , CT~,,,__~, . . . , are the reversed Pad& denominators on the main diagonal in the Pad& table for g(z) =

exp(2C”,=,a2m_,z2”-1/(2m

- 1)).

Proof. Let n denote a fixed but arbitrary integer > 1. 2”-1/(2m - 1)); then, differentiating, We put h(z) = exp(2C;=,a2,_,z

2 2 u2m- 1’ 2m-2 h(z). 1 i m=l From this it follows that h(O) = 1, h’(O) = 2~7,. For g(z) we have h’(z)=

g’(z)

=

(1.3)

UZm-1Z

with g(0) = 1, g’(0) = 2 ur. Differentiating

both members in (1.3) k times, we obtain

g(k+r)(Z) =

I

2m-2

xi

@2m-1Z

m=l

+

2k I(

m=l

u2m-1Z

2m-2

+ I

O(z2”)

1

g’“‘(z).

+

0(~2”-1

)I

g(k-l)(z)

(14

H. Van Rossum / Polynomial sequences with prescribed zeros

319

If we delete O(Z~~), O(Z~~-~), . . . ,O(Z~~-~ ) from (1.4), we obtain the expression for hCk+‘j(z). Hence, as long as k < 2n, we have gCk)(0) = /Z(~)(O).The MacLaurin expansions of g(z) and h(z) agree up till the term in z2” inclusive. But then the main diagonal sequences of Pad& approximants to g(z) and h(z) agree up till the nth approximants inclusive. Specifically, the PadC approximant on (n, n) in the table for g(z) is equal to the one on (n, n) in the table for h(z). Since h(z) =f(z-‘>, wh ere f(z) is given by (1.21, the proof is completed. q In the remainder of this paper a major role is played by a corollary to Theorem 1.2 which we record here as the following proposition. Proposition 1.3. The polynomials V,(z), m = 1, 2,. . . , with prescribed psz (u2,,, _ ,I”, = 1, form an orthogonal polynomial system COPS) with respect to the sequence (c,)~,~, where C~,Oc,~m is the MacLaurin expansion of exp[2C”, = iu2,,, _ 1z 2m- ‘/(2m - 111.

2. Case I

In this section we consider the case where only a finite number of the prescribed psz differs from zero. Example 2.1. The prescribed psz are (hi = i, g2,,_ i = 0, m = 2, 3,. . . . The polynomials V,(z) with these psz are the reversed Pade denominators on the main diagonal in the table for exp(z). They are orthogonal with respect to the sequence 1, (2!)-‘, (3!)-‘, . . . . We will work here, however, with the Pad& numerators U,(z), where U,(z) = V,< -z), m=1,2,... . The system (z~U~(Z-~)>~=, has psz - 3, 0, 0,. . . . This is an OPS with respect to the sequence -1, (2!)-l, -(3!>-l, (4!)-l,. . . . Apart from the substitution z + 22, this is the OPS of ordinary Bessel polynomials (see [4]). For the ordinary Bessel polynomials we have psz - 1, 0, 0,. . . . This property of ordinary Bessel polynomials to be (in our formulation) a system of polynomials with prescribed psz has been known (implicitly) for a long time. Grosswald [4] refers to Burchnall [2]; actually, the result was established by Laguerre [6, pp. 117, 1181 in 1880. Example 2.2. The prescribed

(pi = +a f 0,

u3=

psz are -7,

3

u~~_..~=O,

m=3,4

,...

.

The OPS (V,
g K(a) m=O m!

Zm*

equation f’(z) + (3z2 - a>f(z) = 0. This can be written as

(2.1)

320

H. Van Rossum / Polynomial sequences with prescribed zeros

Equating to zero the coefficient of z” in (2.11, we obtain the recurrence H,+,(a)

- &?,(a)

+ 3m(m - l)H,_*(a)

= 0,

relations

m = 2, 3,. . .)

(2.2)

initialized by H,(a) = 1, H,(a) = a, H,(a) = u2. From this we see that the moments c,,, = H&)/m!, m = 1, 2,. . .) of the OPS (V,( ~>>“,=aare polynomials in a. They are closely related to the polynomial systems considered earlier in [3,7]. It is clear that the parameter a can be chosen as to ensure the normality of the sequence (H,(u)/m!)“,=,. We can obviously state that the main diagonal in the Pad6 table for exp[2Ck= 1~2m_,z2”-‘/(2m - l)] is normal, apart, eventually, from a countably infinite number of exceptions. The requirement (3) in Definition 1.1 is not too restrictive. Therefore, in the remainder of this section we assume normality of the main diagonals involved. We will deal with Case I now in its general form. So, let P,(z)/&,(z) denote the (m, m)-Pad6 approximant to g(z), given as ,

n

a fixed integer,

(T~~_~+ 0.

(2.3)

The Laguerre-Perron theory (see [8, pp. 250 ff) yields a linear second-order differential equation with polynomial coefficients for the Pad6 denominators Q,,,(z), m = 1, 2,. . . , of g(z). Perron’s differential equation becomes in our case (writing Q, instead of Q,,,> zO,QII, - [ Mzf9, + ztl,‘,, + 2m0,] Ql, + H,Q, where M= -a,-u3z2satisfying the differential Q,P;

- P,Q;

*.* -U2n-1Z equation +

2n-2,

H,

= 0,

is a polynomial

(24

and em is a polynomial

MQ,P, = Z2*em.

(2.5)

It is clear that deg 0, = deg M = 2n - 2. Theorem 2.3. If an OPS is a polynomial sequence with prescribed psz, then every member of the OPS satisfies a linear second-order differential equation with polynomial coefficients that depend on m, but which are of bounded degree. Proof. Let (V,( z))z=-, be the OPS with psz ul, (us,. . . , u2n_ 1, where n is a fixed integer > 1 and (T~~_~#O, u~,,_~=O, m=n+l,n+2 ,.... Put V,=Z~Q~(Z-~), m=O,l,.... Now

Q,(Z) is the mth-degree

polynomial appearing in (2.4). From this the assertion follows.

Theorem 2.4. Let LY# 0 be a zero of Q,(z), diagonal in the Pude’ table for

the Pude’ denominator of degree m on the main

1 exp 2 ulz + $r,z3 + - *. + ~ ii

Assume e,(o) zero of Q,(z).

2n - 1

0

u22n-1=

2n-1 II

# 0, where 8, is the polynomial appearing in (2.4) and (2.5). Then a is a simple

321

H. Van Rossum / Polynomial sequences with prescribed zeros

Proof. By contradiction. Assume Q;<(Y) = 0. From (2.4) it follows differentiating, the left-hand member of (2.4) becomes (ze,)‘Q;

+ z&Q;

- [ MzO,

that

Q$<(-Y>= 0. Upon

+ ~0; + 2me,]‘Q:,

-[Mze,+ze:,+2me,]Q~+H~Q,+H,Q:,, from which it follows that Q:(a) = 0. Repeated application yields that all derivatives to the mth derivative inclusive, are zero at z = (Y. This is impossible. 0

of Q,,

up

Theorem 2.5. Zf (Y # 0 is a common zero of Q, and 8, (see (2.5)), then (Y is a multiple zero of

Q ??I’ Proof. Q,Ja> = 0,((~> = 0. Then from (2.5) it follows that P’(cY)&~((Y) = 0. Since P,(z)/Q,(z) is irreducible, P,(a) # 0, hence QL = 0. 0 The results in Theorems 2.4 and 2.5 can be translated immediately into results for the zeros of the members of the OPS with prescribed psz (or, ~~,...,u~~_~ ~0, m =n + 1, n +2,... . For instance, parallel to Theorem 2.4 we have the following theorem. Theorem 2.6. Let /3 # 0 be a zero of the orthogonal polynomial V,(z) = zmQm(z-‘1, defined as in Theorem 2.4. Zf 9(/3 - ‘># 0, then p is a simple zero of V,.

where Q, is

Remark 2.7. From Theorems 2.4 and 2.5 (or 2.6) it follows that if V~ is the number of multiple zeros of V, and p,,, is the number of simple zeros of V,, we have v,/~~ + 0 if m -+ 03. Remark 2.8. If the polynomial ( cl7

u3~...,u2~-_1,

sequences 0,

0,.

..)

(V$)< z >I”,= 0 and ( Vi2)Cz>>“,= 0 possess psz respectively and

(~1,

~33,...,~2~-1,

~2~+1,

~0, then VL1)(z)=VL2’(z) if m=O, l,..., n. For the moment sequences of these OPS, (c:)>“,=~ and (c:))“,=~ c(i) = c(2) m, m =O, l,..., 2n. m where

0,

O,...),

u22n- 1@2, + 1

respectively,

we have

3. Case II In the sequence

of prescribed

g(z)=exp

2 iuZn_rz L

n=l

The nth PadC approximant h(z) = exp 2 t [

compare Theorem true for Theorem

psz (uZn_ ,)I=,,

m=l

to g(z) u2m_l g

1 1

infinitely

many terms differ from zero. We put

.

is equal to the nth one to h(z), where ,

1.2. So it is clear that Theorems 2.4 and 2.5 carry over to Case II. This is not 2.3, but we do have the following theorem.

322

H. Van Rossum / Polynomial sequences with prescribed zeros

Theorem 3.1. The OPS withprescribedpsz ((T~~_~)~=~, and where C~=1~2n_1z2n-1/(2n - 1) has a rational derivative, has the property that every member of the OPS satisfies a linear second-order differential equation with polynomial coefficients of bounded degree.

Example3.2.Thepszarea2,_,=o,n=1,2,..., consists of the reversed Pad& denominators i.e., g(z,

O<]o]
+ +z3 + +z5 + . . * )) = (p$

co) = exp(o(z

O

oER.

This holds for all z E C with ( z I < 1. The polynomials are PA-“+‘)(z), m = 0, 1,..., where P(@), cy> - 1,p > - 1,is the Jacobi polynomial of degree m. See [6, pp. 344-3591. m on the table for g(z, o) are PC*+)(z), m = 0, 1,... . For, because of g( -2, o) = (g(z, w>>-‘, the reversed numerators R,(z) satisfy R,(z) = P;-‘,‘)( -z), m = 1,2, . . . ) but then from g(z, w) = g( -z, -w) it follows R,(z) = P(w9-w)(z), m = 1,2,... . Remark 3.3. The reversed numerators

Remark 3.4. For no other specification

of the parameters

(Y,p, (P~‘@‘(z))“,=~ is an OPS with

prescribed psz. Indeed,

P$“‘P’(Z) = +( [(a + p)’ + 7(0! + p) + 12122 + 2(a - P)(a + P + 3)” +

(a -

p)’ - (cr + p - 4)).

The sums of the zeros of P, and P, are, respectively, -2(&! - @)(a + p + 3)

-CY+p (Y+p+2’

(,+~)“+7((~+~)+12’

(3.1)

If (Y= p, the sums are equal, but in this case we have the ultraspherical polynomials and then all odd psz are nought. This is precluded by Definition 1.1. Now let (Y# p. Equating the sums in (3.1) and dividing by cr - p, we find (Y+ /I = 0 or (Y= -p as a necessary condition. It is also sufficient, as we have seen. In the same manner one can check that the Laguerre nor the Hermite polynomials belong to the class of OPS with prescribed psz.

4. Totally positive polynomials

In a number of papers, Schoenberg [9] and Arms and Edrei [l] studied meromorphic functions, representable in the form f(z)

= coeyz fp (l+~jZ)(l-pjZ)-', j=l

a class S of

323

H. Van Rossum / Polynomial sequences with prescribed zeros

Let Ci =Oc, 2”’ be the MacLaurin expansion of f. In [l] it is proven that if f E S is not rational, then the following holds. (1) The Pad& table for f is normal; specifically, all Toeplitz determinants of the sequence (c,)“,=, are positive. A consequence is firstly c, > 0, m = 0, 1,. . . , and secondly “‘>“‘>““> co Cl

. ..* c2

(2) The sequence of numerators and the sequence of denominators the Pade table for f converge separately, respectively to coey’/2 fi (1 + ajz)

and

e-yr/2 IFI (1 j=l

j=l

on the main diagonal in

-Pjz)*

(4.1)

The convergence is uniform in any bounded domain of the complex z-plane. Remark 4.1. A stronger statement

than the one contained in (2) was actually proven in [l]. The in (2) is sufficient for our purposes in this paper.

result as formulated

In the remainder of this paper we consider a subclass T c S of meromorphic representable in the form aj>Ofor

f(~)=e~fi(l+ru~~)(l-~~~t)-‘,

functions f,

all j> 1.

j=l

4.2. The polynomials zmUm(z-‘>, m = 0, 1,. . . , where U,
c

is the (m, m)

j=l

m=O

(4.2)

and where aj 2 0, for all j = 1, 2,. . . , are called totally positive polynomials (TPP). They were studied by the author in 1965, see [lo]. Remark 4.3. The ordinary Bessel polynomials

are obtained by putting

(4.2). Theorem 4.4. The sequence of TPP is a sequence with prescribed psz.

Proof. Using (4.21, we have (for ( z ( small enough) f(Z)=eW

~+l~~,~~(l+~jz)-l~~flI(l~~jz) [

1 +cXjz = exp z + 2 log 1 -Crjz I j=l [ -CXP{2[(+

+ ilaj)Z+

i(FlaT)Z3+

‘--I;.

‘Ye= 0, j = 1, 2, . . ., in

324

H. Van Rossum / Polynomial sequences with prescribed zeros

Let U,(z)/V,(z) denote the (m, m) Pad& approximant to f; then, recalling U,(z) = V,(-zl, we see that the sequence (z~U,
&Xj,

- &r;,

j=l

- &Xl )....

j=l

0

j=l

We notice the following properties of the TPP. (I) The coefficients are all positive. (II) The sequence (zmUm(z-‘>>~=, is orthogonal with respect to the sequence -ci, c2, -c3, CJ, -cg )... . (III) The TPP satisfy linear-second order differential equations with polynomial coefficients. Remark

4.5. The ordinary Bessel polynomials are the reversed numerators on the main diagonal of the Pade table for exp(z) = iF,(l; 1; z). Krall and Frink’s generalized Bessel polynomials are connected with the Pad& table for ,F,(l; c; z), c # {O, - 1, - 2,, . . }, so in this generalization the hypergeometric character of the ordinary Bessel polynomials is retained. In our generalization of the ordinary Bessel polynomial sequence, the property of forming a sequence with prescribed psz is retained.

We will now give several results on the zeros of the TPP. Theorem 4.6. The zeros of the TPP in Definition 4.2 are on the disc ) z ) < cJc,. Remark 4.7. A weaker

form was proved by the author [lo], where the bound cl/cO was established. The proof given there can easily be modified so as to yield Theorem 4.6.

Theorem 4.8. The zeros of the TPP z~&(z-~), m = 0, 1, 2 ,..., -c3, * . . .) where C~,O~,~m is the MacLaurin expansion of f(z)

= exp(z),fi

(~~20,

z7

forj=

with moments

-cl,

c2,

1,2 ,...,

are all on the left half of the disc 1z ) < c2/cI. The zero of the first-degree polynomial is - icl and it lies on the boundary of the disc (showing the bound c2/cI to be sharp). Proof. For the (m, ml-Pad6 approximant U,(z)

= 1 + a,z + -. . +a,zm,

to f(z),

i.e., U,(z>/V,,

Vm(z) = 1 -aa,z+

we have

0.. +(-l)mamzm,

( 1 - a,z + . . * + (- l)mamzm)f(z) .* . +a,z”)=O(z2m+1). -(I +a,z+ From this a, = kc1 follows. Since (z~U,(Z-‘>>“,=~ is a sequence with prescribed psz, we have U;(O) = - $cl, m = 1, 2,. . . . The recurrence relations for three consecutive Pad& numerators on the main diagonal are [8, pp. 260, 2611 U,(z)

Moreover,

= (1 + lmz)Um_l(z)

1, = U;(O) - U;_,(O),

+ k,~~U~_~(z),

m = 1, 2,. . . , whence

m = 2,3..

. .

P-3)

1, = 0, m = 2, 3.. . . Also we have I1

H. Van Rossum / Polynomial sequences with prescribed zeros

325

Fig.l.z=x+iy,i=m.

= icl. From a result in [l], it follows that k, > 0, m = 2, 3,. . . . The (4.3) are to be initialized by U,(z) = 1 + $rz, U&z> = 1. A result of [ill implies that with these conditions the recurrence polynomials whose zeros are all in the open left half-plane. This holds zmUm(z-l), m = 1, 2 )... . Hence appealing to Theorem 4.6, the first proved. Now from (4.2) we find, by logarithmic differentiation,

fYz) =

recurrence

relations in

relations in (4.3) yield also for the polynomials part of the theorem is

l+

f(z)

Since c0 = f(O) = 1, we find c1 = f ‘(0) = 1 + 2C’7’=,crj. Differentiating f ‘(z>/f(z>, we get f “(z>f(z)

- (f’(zH2

(f(z)>”

= _ e j=l

a;

(1 +

+ 2

CljZ)’

Putting z = 0, we find f “(0) = c:, whence c2/cI = $r.

a;

j=l

(1

-CijZ)’

*

0

Corollary 4.9. The nonreal zeros of the TPP are all in the intersection {z 1- icl < Re z < 0) n {z II z ( =G+c,>.

326

H. Van Rossum / Polynomial sequences with prescribed zeros

Proof. We pick a certain TPP as in Definition

4.2. Since all its coefficients are real, nonreal zeros can occur only in pairs of conjugate complex numbers. Let z0 be such a zero and Im z0 # 0. Since the sum of the zeros of the polynomials is equal to - icl, we have - icl 0, there exists an N, such that f,,,(z) f or m > N, has the same number of zeros in thedisc Iz--al
For this version of the theorem and its proof, compare [5, p.2051. Theorem 4.11. Let f(z) = (exp(z))IITZ1,(l + (~~z)(l - ajt)-’ and o1 > (Ye > * * * > 0. Let Cy=,oj < co and let Cm,=O~,~m denote the MacLaurin expansion off(z). Furthermore, let U,(z) be the (m, m)-Pad& numerator in the table for f(z). Assertion: each - l/(rj, j = 1, 2,. . . , is an accumulation point of real zeros of U,(z), m = 1, 2,. . . . Proof. Let us pick a certain aj. Hurwitz’ theorem applies, i.e., for every E > 0, there exists an NE

such that m > N,

*

the disc Iz + l/cujI
Let z0 be such a zero; then z0 E [w,since, if Im z0 # 0, also Z, (another zero of U,(z)) would be on the disc. This is impossible. The proof is completed as soon as we have ruled out the possibility that all U,(z) from a certain index m, on would vanish at - l/aj. For assume U,( - l/cuj> = 0, m = m,, m, + 1,. . . . Then from the recurrence relations (4.3) it follows that U,(- l/cuj> = 0. However, U,(z) = 1 + (tc,>z. 0 An immediate

corollary to Theorem 4.11 is the following theorem.

Theorem 4.12. The accumulation points of the (real) zeros of the TPP belonging to the moment generating function f(z) in Theorem 4.11 are -oj, j = 1, 2,. . . .

A corollary to Hurwitz’ theorem, applied to Theorem 4.11, gives the statement that - l/aj, j=l,2 , . . . , are the only accumulation points of zeros. In the transition from Theorem 4.11 to Theorem 4.12, a transformation t + z ~ ’ is involved, hence the statement that --ai, j = 1, 2,. . . , are the only accumulation points of zeros of the TPP is restricted to domains in C\(O). Theorem 4.13. The origin of the complex plane is an accumulation point of zeros of TPP. Proof. Case I. Infinitely many ayi differ from zero. Then the origin is an accumulation

accumulation

points of zeros, so is itself an accumulation

point of zeros of the TPP.

point of

H. Van Rossum / Polynomial sequences with prescribed zeros

327

Case II. Let (hi > (Ye> . . . > (Ye, ‘Y~= 0, j =p + 1, p + 2,. . . . Let B,(z) denote an arbitrary but fixed TPP with zeros tim), z$~), . . . , ~2). Then we have C’/+Z~~~ = - +cl. Let z’,“) be one of the zeros with smallest absolute value of its real part. Then we have 1Re z’,“‘I < c,/(2m). So we can build a well-defined sequence (1Re z~‘[>~=~ converging to 0. So a strip

{z~-~~Rez
a,>e>O,

contains infinitely many zeros of the sequence of the TPP. There must be at least one q accumulation point. Such a point cannot belong to C\{O). Hence, 0 is the only possibility. If ‘yj = 0, j = 1, 2,. ..) in Definition 4.2 of the TPP, they reduce to ordinary polynomials. The following well-known result follows from Theorems 4.12 and 4.13.

Bessel

Theorem 4.14. The origin of the complex plane is the only accumulation point of zeros of the ordinary Bessel polynomials.

References [l] R.J. Arms and A. Edrei, The Pade tables and continued fractions generated by totally positive sequences, in: Mathematical Essays Dedicated to A.J. Ma&tire (Ohio Univ. Press, Athens, OH, 1970) 1-21. [2] J. Burchnall, The Bessel polynomials, Canad. .Z.Math. 3 (1951) 62-68. [3] N.E.F. de Battig, Nota sobre polinomios generalizados de Hermite, Reu. Ser. A Mat. Fys. Teor. Uniu. National Tucuman 24 (l-2) (1974) 247-249. [4] E. Grosswald, Bessel Polynomials, Lecture Notes in Math. 698 (Springer, Berlin, 1978). [S] E. Hille, Analytic Function Theory, Vol. ZZ(Ginn, Boston, MA, 1962). [6] E. Laguerre, Oeuures, Tome Z (Chelsea, New York, 1972). [7] M. Lahiri, On a generalisation of Hermite polynomials, Proc. Amer. Math. Sot. 27 (1) (1971) 117-121. 181 0. Perron, Die Lehre uon den Kettenbriicken, Band ZZ(Teubner, Stuttgart, 1957). [9] I.J. Schoenberg, Uber variationsvermindernde lineare Transformationen, Math. Z. 32 (1930) 321-328. [lo] H. van Rossum, Totally positive polynomials, Proc. Kon. Nederl. Akud. Wetensch. Ser. A 68 (2) (1965); Zndag. Math. 27 (2) (1965) 305-315. [ll] H.S. Wall, Polynomials whose zeros have negative real parts, Amer. Math. Monthly 52 (1945) 308-322.