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On the real-rootedness of generalized Touchard polynomials
Applied Mathematics and Computation 254 (2015) 204–209
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On the real-rootedness of generalized Touchard polynomials Chak-On Chow a, Toufik Mansour b,⇑ a b
Division of Science and Technology, BNU-HKBU United International College, Zhuhai 519085, PR China Department of Mathematics, University of Haifa, 3498838 Haifa, Israel
a r t i c l e
i n f o
a b s t r a c t
Keywords: Generalized Touchard polynomials Real-rootedness Interlacing Alternating
We consider the real-rootedness of generalized Touchard polynomials recently revisited by Mansour and Schork (2013). Towards this end, we first describe the normal form of the generalized Touchard polynomials, by which recurrence relations for the polynomial part are derived. By using the recurrence relations, we prove the real-rootedness of the generaln o k :k2N . ized Touchard polynomials for the parameter m 2 ½1; 1Þ [ kþ1 Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Jacques Touchard [17,18] introduced in 1933 the Touchard polynomials T n ðxÞ’s, in the study of permutations with cycles satisfying certain conditions, as extensions of the partial Bell polynomials [1]. The Touchard polynomials T n ðxÞ’s (also known as the exponential polynomial or Bell polynomial) can either be defined by the exponential generating function P P P j n tn xðet 1Þ or by the Stirling transform T n ðxÞ ¼ nj¼0 Sn;j xj , where Sn;j ¼ 1j! j‘¼0 ð1Þj‘ ‘ denotes the Stirling nP0 T n ðxÞ n! ¼ e ‘ number of the second kind, which counts the number of partitions of an n-set into j non-empty blocks (see [12]). It is well known that the moments of the Poisson distribution are intimately related to the combinatorics of Stirling numbers of the second kind and Bell numbers. More precisely, if the random variable X PoissonðkÞ, then for n ¼ 1; 2; . . ., the Touchard polynomial T n ðxÞ satisfies the relation
T n ðkÞ ¼ E½X n ; i.e., T n ðkÞ is the nth moment of X. Moreover, several properties of Touchard polynomials are studied in [3,5,10,18], where some of them are applied to problems in random walks. Touchard polynomials have been extended in many contexts, see [4,5,10,14–16] for instance. Of relevance to the present work is the higher order extension of Touchard polynomials introduced by Dattoli et al. [7], defined for m 2 Z; n 2 N by n
T nðmÞ ðxÞ :¼ ex ðxm @ x Þ ex ;
ð1:1Þ
which reduce when m ¼ 1 to the classical Touchard polynomials T n ðxÞ mentioned above. Dattoli et al. [7] discussed several properties of these polynomials, including the recurrence (1.2) below for m 2 N (also, see [11]). Mansour and Schork [13] ðmÞ recently revisited T ðmÞ n ðxÞ by extending m to arbitrary real number and showed various properties of T n ðxÞ’s, including the following recurrence relation
⇑ Corresponding author. E-mail addresses: [email protected] (C.-O. Chow), [email protected] (T. Mansour). http://dx.doi.org/10.1016/j.amc.2014.12.108 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
C.-O. Chow, T. Mansour / Applied Mathematics and Computation 254 (2015) 204–209 ðmÞ
ðxm þ xm @ x ÞT ðmÞ n ðxÞ ¼ T nþ1 ðxÞ
205
ð1:2Þ
ð12Þ
as well as an identity relating T n ðxÞ and the nth Hermite polynomial Hn ðxÞ. See (4.1) in Section 4. Since Hn ðxÞ possesses the ð1Þ remarkable property of being real-rooted, this suggests that T n2 ðxÞ enjoys this remarkable property also. A crucial property ð1Þ ð1Þ ð1Þ of T n2 ðxÞ not explored by Mansour and Schork is the representation T n2 ðxÞ ¼ xan T~ n2 ðxÞ, where an 2 0; 12 and ð1Þ ð1Þ ð1Þ T~ n2 2 Qþ ½x. Since the real-rootedness of T n2 ðxÞ implies that of T~ n2 ðxÞ, an application of the Aissen–Schoenberg–Whitney P ð1Þ theorem [2, Theorem 2.2.4] to T~ n2 ðxÞ ¼ di¼0 ai xi then yields the total positivity consequence that any minor of the infinite ð1Þ matrix M ¼ ðM ij Þi;j2N defined by M ij ¼ aji for all i; j 2 N (where ak ¼ 0 if k < 0 or k > d ¼ deg T~ n2 ðxÞ) is non-negative. Because of these, it is of considerable interest to study more generally the real-rootedness of T ðmÞ n ðxÞ for m 2 R. The organization of this paper is as follows. In the next section, we consider the positive integral m case. In Section 3, we consider the negative integral m case. In Section 4, we study how the positive integral m case extends to give the positive real m case. 2. The positive integer case We study in the present section the real-rootedness of T ðmÞ n ðxÞ for m 2 N. Let m 2 N. One can compute T ðmÞ n ðxÞ either directly by (1.1), or by the recurrence (1.2). The first few members are listed as follows: ðmÞ
T 1 ðxÞ ¼ xm ; ðmÞ
T 2 ðxÞ ¼ x2m1 ðx þ mÞ; ðmÞ
T 3 ðxÞ ¼ x3m2 ½x2 þ 3mx þ mð2m 1Þ; ðmÞ
T 4 ðxÞ ¼ x4m3 ½x3 þ 6mx2 þ mð11m 4Þx þ mð2m 1Þð3m 2Þ; ðmÞ
T 5 ðxÞ ¼ x5m4 ½x4 þ 10mx3 þ 5mð7m 2Þx2 þ 5mð2m 1Þð5m 2Þx þ mð2m 1Þð3m 2Þð4m 3Þ: We see that when m 2 N, all coefficients of T ðmÞ n ðxÞ are positive integral. ðmÞ
Proposition 1. For any positive integers m and k; T k ðxÞ is a monic polynomial of degree km with non-negative integer coefficients.
Proof. We fix the positive integer m and proceed by induction on k. Since ðmÞ
T 1 ðxÞ ¼ ex ðxm @ x Þex ¼ xm ; the theorem clearly holds when k ¼ 1. Assume that the result holds up to k. By the recurrence relation ðmÞ
ðmÞ
ðmÞ
ðmÞ 0
T kþ1 ðxÞ ¼ ðxm þ xm @ x ÞT k ðxÞ ¼ xm T k ðxÞ þ xm ðT k Þ ðxÞ; which is a monic polynomial of degree m þ km ¼ ðk þ 1Þm.
h
It is evident from the above list that
T nðmÞ ðxÞ ¼ xnmnþ1 T~ ðmÞ n ðxÞ
ð2:1Þ
ðmÞ ðmÞ for some monic polynomial T~ n 2 N½x with T~ n ð0Þ > 0. We shall show that a sufficient condition for the coefficients of ðmÞ ~ T n ðxÞ to be positive integral for all n is that m 2 N.
ðmÞ Proposition 2. For n 2 N, the Touchard polynomial T~ n ðxÞ is monic, and satisfies the recurrence relation
0 ðmÞ ðxÞ: T~ nþ1 ðxÞ ¼ ðx þ nðm 1Þ þ 1ÞT~ nðmÞ ðxÞ þ x T~ ðmÞ n If m 2 N, then
ðmÞ T~ n
ð2:2Þ
2 N½x.
Proof. Substituting (2.1) into (1.2), we have ðmÞ
0
~ ðmÞ xðnþ1Þmn T~ nþ1 ðxÞ ¼ ðxm þ xm @ x Þxnmnþ1 T~ nðmÞ ðxÞ ¼ xðnþ1Þmn ½ðx þ nm n þ 1ÞT~ ðmÞ n ðxÞ þ xðT n Þ ðxÞ: Cancelling xðnþ1Þmn from both sides, (2.2) follows. The multiplicative factor x þ nm n þ 1 has positive coefficients () m > 1 1n. Thus, for m 2 N, the latter condition holds and both factors on the right side of (2.2) have positive integral ðmÞ coefficients; T~ n 2 N½x then follows by induction. h
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We also need some notations from [8] concerning the relative positions of real zeros of polynomials. A real polynomial f ðxÞ is standard if its leading coefficient is positive; it is simply real-rooted if all its zeros are real and simple. Let f ðxÞ and gðxÞ be simply real-rooted standard polynomials of degrees n and m, respectively, and let h1 < h2 < < hn and n1 < n2 < < nm be their zeros. Then gðxÞ is said to interlace f ðxÞ if n ¼ m þ 1 and
h1 6 n1 6 h2 6 n2 6 6 hn1 6 nn1 6 hn ; gðxÞ is said to alternate left of f ðxÞ if n ¼ m and
n1 6 h1 6 n2 6 h2 6 6 nn 6 hn : In case all the above inequalities are strict, we say that gðxÞ strictly interlaces (resp., strictly alternates left of) f ðxÞ. It is easy to see that if x 2 ðhi ; hiþ1 Þ, then sgnf ðxÞ ¼ ð1Þni , where i ¼ 0; 1; . . . ; n and h0 :¼ 1. This fact will be exploited when proving Theorems 3 and 6. Theorem 3. For any positive integers n and m; T ðmÞ n ðxÞ has the trivial zero x ¼ 0 of multiplicity nm n þ 1, and n 1 simple ðmÞ ðmÞ negative zeros. Moreover, T~ n ðxÞ strictly interlaces T~ nþ1 ðxÞ for n P 2. Proof. We fix the positive integer m and proceed by induction on n, the case n ¼ 1 being trivial by virtue of Proposition 1. ðmÞ Since T~ 2 ðxÞ ¼ 0 () x ¼ m, it follows that ðmÞ T~ 3 ðmÞ ¼ ðmÞ2 þ 3mðmÞ þ mð2m 1Þ ¼ m < 0: ðmÞ ðmÞ Since the graph of T~ 3 ðxÞ is a parabola opening upward, T~ 3 ðmÞ < 0 implies that m must lie in between the zeros of ðmÞ ðmÞ i.e., T~ ðxÞ strictly interlaces T~ ðxÞ. Assume that the result holds for n P 3. Let xn;1 < xn;2 < < xn;n1 < 0 be the
ðmÞ T~ 3 ðxÞ,
2
3
simple zeros of T ðmÞ n ðxÞ. Define also xn;0 :¼ 1 and xn;n :¼ 0. By the recurrence relation (2.2), we have that 0 ðmÞ T~ nþ1 ðxÞ ¼ ðx þ nm n þ 1ÞT~ nðmÞ ðxÞ þ xðT~ ðmÞ n Þ ðxÞ:
Upon setting x ¼ xn;j ; j ¼ 1; 2; . . . ; n 1,
0 ðmÞ T~ nþ1 ðxn;j Þ ¼ xn;j T~ nðmÞ ðxn;j Þ so that ðmÞ sgnT~ nþ1 ðxn;j Þ ¼ ð1Þnj :
ðÞ
ðmÞ ðmÞ Since deg T~ nþ1 ðxÞ ¼ n; sgnT~ nþ1 ðxn;0 Þ ¼ ð1Þn . Since
ðmÞ T~ nþ1 ðxn;n Þ ¼ ðnm n þ 1ÞT~ ðmÞ n ðxn;n Þ > 0; ðmÞ
sgnT~ nþ1 ðxn;n Þ ¼ 1. Thus, (⁄) actually holds for j ¼ 0; 1; . . . ; n. So, for j ¼ 1; 2; . . . ; n, there exists xnþ1;j 2 ðxn;j1 ; xn;j Þ such that ðmÞ ðmÞ ðmÞ ðmÞ T~ nþ1 ðxnþ1;j Þ ¼ 0. These account for those n simple negative zeros of T nþ1 ðxÞ. Since T nþ1 ðxÞ ¼ xðnþ1Þmn T~ nþ1 ðxÞ; x ¼ 0 is a zero ðmÞ
of T nþ1 ðxÞ of multiplicity ðn þ 1Þm n. This finishes the induction and the proof of the theorem.
h
The same idea of proof has been employed by Chow [6] to prove the real-rootedness of the generating function dn ðxÞ of multiderangements by excedances. 3. The negative integer case When m is negative, we shall write T nðmÞ ðxÞ with m 2 N. Mansour and Schork [13] showed that the following recurrence relation ðmÞ
T nþ1 ðxÞ ¼ ðxm þ xm @ x ÞT ðmÞ ðxÞ n T nðmÞ ðxÞ.
holds for A comparison of (1.2) and (3.1) shows that the latter is the former with m in place of m. Let us first inspect the first few members of T nðmÞ ðxÞ, as follows: ðmÞ
T1
ðxÞ ¼ xm ;
ðmÞ T 2 ðxÞ ðmÞ T 3 ðxÞ ðmÞ T 4 ðxÞ ðmÞ T 5 ðxÞ
¼ x2m ð1 mx1 Þ; ¼ x3m ½1 3mx1 þ mð2m þ 1Þx2 ; ¼ x4m ½1 6mx1 þ mð11m þ 4Þx2 mð2m þ 1Þð3m þ 2Þx3 ; ¼ x5m ½1 10mx1 þ 5mð7m þ 2Þx2 5mð2m þ 1Þð5m þ 2Þx3 þ mð2m þ 1Þð3m þ 2Þð4m þ 3Þx4 :
ð3:1Þ
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The above listed members suggest that
T ðmÞ ðxÞ ¼ xnm f n ðx1 Þ; n
ð3:2Þ
where f n 2 Z½x satisfies f n ð0Þ ¼ 1; deg f n ðxÞ ¼ n 1, and the coefficients of f n ðxÞ alternate in signs. Thus, the polynomial f n ðxÞ has all its coefficients positive integral. Proposition 4. For m; n 2 N,
T ðmÞ ðxÞ ¼ xnm f n ðx1 Þ; n where f n 2 Z½x is of degree n 1 and f n ð0Þ ¼ 1. Moreover; g n 2 N½x, where g n ðxÞ :¼ f n ðxÞ. Proof. From the recurrence relation (3.1) and (3.2), we have
xðnþ1Þm f nþ1 ðx1 Þ ¼ ðxm þ xm @ x Þxnm f n ðx1 Þ:
ð3:3Þ
Let y ¼ x1 and g n ðyÞ ¼ f n ðyÞ. Then @ x ¼ dy @ ¼ y2 @ y so that (3.3) becomes dx y
ðyÞðnþ1Þm g nþ1 ðyÞ ¼ ððyÞm þ ðyÞm y2 @ y ÞðyÞnm g n ðyÞ ¼ ðyÞðnþ1Þm g n ðyÞ þ ðyÞm y2 ½ð1Þnm nmynm1 g n ðyÞ þ ðyÞnm g 0n ðyÞ ¼ ðyÞðnþ1Þm ½ð1 þ nmyÞg n ðyÞ þ y2 g 0n ðyÞ: Cancelling ðyÞðnþ1Þm from both sides, we obtain the recurrence relation for g n ðyÞ:
g nþ1 ðyÞ ¼ ð1 þ nmyÞg n ðyÞ þ y2 g 0n ðyÞ: Since g 1 ðyÞ 1, it follows from induction that g n ðyÞ has positive integral coefficients only and g n ð0Þ ¼ 1.
h
With the factorization of T nðmÞ ðxÞ as in Proposition 4, one is tempted to believe that T nðmÞ ðxÞ enjoys a similar real-rootedðmÞ
ðxÞ, it is readily seen that T 2 ness. Unfortunately, this is not the case. From the above list of T ðmÞ n root x ¼ m for any m P 1. But
ð1Þ T 3 ðxÞ
5
ðxÞ ¼ 0 has the only real
2
¼ x ðx 3x þ 3Þ ¼ 0 has non-real roots. Computer evidence further reveals that
ðxÞ ¼ 0 has only real roots, which do not exhibit the usual interlacing property with respect to those when m P 3n; T ðmÞ n ðmÞ
of T nþ1 ðxÞ ¼ 0. Because of the latter observation, we shall not dwell on this case any further. 4. The positive real case With the real-rootedness of T ðmÞ n ðxÞ established in the affirmative in the positive integral m case as in Section 2, we would like to see whether the same is true when m ¼ a 2 Rþ . A closer examination of the proof of Theorem 3 reveals that if the coefficients of T ðmÞ n ðxÞ are positive, not necessarily integral, then the real-rootedness of T ðmÞ n ðxÞ for m P 1 continues to hold, should m be integral or not. It remains to consider the remaining m 2 ð0; 1Þ case, thus completing our discussion of the positive m case. Mansour and Schork [13, Theorem 5.3] showed that
ð1Þ T n2 ðxÞ ¼
n pffiffiffi
i Hn i x ; 2
where Hn ðxÞ is the nth Hermite polynomial defined by the generating function identity: 2
e2tzt ¼
X
Hn ðzÞ
nP0
tn : n!
An alternative way to define the Hermite polynomials is by the classical Rodrigues formula:
Hn ðxÞ ¼ ð1Þn ex
2
n d 2 ex : dx
It is well known that Hn ðxÞ is real-rooted. From (4.1), we see that
ð1Þ T n2 ðx2 Þ ¼
n qffiffiffiffiffiffiffiffiffi i Hn i x2 ¼ 0 2
for any root x 2 R of Hn ðxÞ ¼ 0. It is this particular case which motivates our study of the m 2 ð0; 1Þ case. ðkÞ Let us first inspect the first few members of T nkþ1 ðxÞ with k ¼ 3, as follows:
ð4:1Þ
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ð 3Þ T 14 ðxÞ ¼ x3=4 ; x2=4 ð 3Þ ð4x þ 3Þ; T 24 ðxÞ ¼ 4 x1=4 ð 3Þ T 34 ðxÞ ¼ ð8x2 þ 18x þ 3Þ; 8 1 ð 3Þ T 44 ðxÞ ¼ ð32x3 þ 144x2 þ 102x þ 3Þ; 32 x3=4 ð 3Þ T 54 ðxÞ ¼ ð32x3 þ 240x2 þ 390x þ 105Þ; 32 x2=4 ð 3Þ ð128x4 þ 1440x3 þ 4200x2 þ 3150x þ 315Þ; T 64 ðxÞ ¼ 128 x1=4 ð 3Þ ð256x5 þ 4032x4 þ 18480x3 þ 27300x2 þ 10080x þ 315Þ; T 74 ðxÞ ¼ 256 1024x6 þ 21504x5 þ 142464x4 þ 349440x3 þ 286020x2 þ 51660x þ 315 ð 3Þ : T 84 ðxÞ ¼ 1024 We see that the fractional exponent of x decreases by steps of 14 and the degree of the polynomial part increases by 1. Once the fractional exponent becomes 0, the fractional exponent of the next member starts from 34 with the degree of polynomial part the same as that of the zero exponent member. The pattern repeats itself. Proposition 5. For k; n 2 N,
ðkÞ ðkÞ T nkþ1 ðxÞ ¼ xðkln Þ=ðkþ1Þ T~ nkþ1 ðxÞ;
ð4:2Þ
ðkÞ where n 1 ln ðmod k þ 1Þ with 0 6 ln 6 k, and T~ nkþ1 2 Qþ ½x is of degree n 1 bn1 c satisfying kþ1
k k 0 k k ln ~ ðkþ1 ðkþ1 Þ Þ ð Þ T n ðxÞ þ x T~ nkþ1 ðxÞ T~ nþ1 ðxÞ ¼ x þ kþ1
ð4:3Þ
when n þ 1X1ðmod k þ 1Þ, and
k 0 k ðkþ1 Þ ðkÞ ð Þ T~ nþ1 ðxÞ ¼ T~ nkþ1 ðxÞ þ T~ nkþ1 ðxÞ
ð4:4Þ
otherwise. ðkÞ ðkÞ Proof. It is evident that (4.2) describes the form of T nkþ1 ðxÞ. It remains to show that T~ nkþ1 ðxÞ satisfies the recurrence relations (4.3) and (4.4). For n þ 1X1ðmod k þ 1Þ; k þ 1jn so that n lnþ1 ðmod k þ 1Þ with 0 < lnþ1 6 k. It follows that n 1 ln ¼ lnþ1 1ðmod k þ 1Þ, where 0 6 ln < k. k Setting m ¼ kþ1 and the representation (4.2) into (1.2), we have that
k 0 k k k ln ðkln Þ=ðkþ1Þ1 ~ ðkþ1 ðkþ1 Þ ðkÞ Þ ð Þ T n ðxÞ þ xðkln Þ=ðkþ1Þ T~ nkþ1 ðxÞ x T nþ1 ðxÞ ¼ xk=ðkþ1Þ xðkln Þ=ðkþ1Þ T~ nkþ1 ðxÞ þ kþ1 k k 0 k ln ~ ðkþ1 Þ ð Þ T n ðxÞ þ x T~ nkþ1 ðxÞ ¼ xðkln 1Þ=ðkþ1Þ x þ kþ1
Comparing this equality with (4.2), (4.3) follows. When n þ 1 1ðmod k þ 1Þ; k þ 1jn so that n 1 k ¼ ln ðmod k þ 1Þ and (4.3) becomes (4.4). Note that all multiplicative factors on the right side of (4.3) and (4.4) have non-negative rational coefficients. Since ðkÞ ðkÞ T 1kþ1 ðxÞ ¼ xk=ðkþ1Þ , it follows by induction that T~ nkþ1 ðxÞ has positive rational coefficients only. k k k ðkþ1Þ ð Þ ðkþ1 Þ ðkÞ ðxÞ ¼ deg T~ nkþ1 ðxÞ þ 1. By (4.4), deg T~ nþ1 ðxÞ ¼ deg T~ nkþ1 ðxÞ. This non-increment of degree occurs By (4.3), deg T~ nþ1 ðkÞ c. h whenever n 1 is a multiple of k þ 1. Consequently, we have deg T~ nkþ1 ðxÞ ¼ n 1 bn1 kþ1 k k ~ ðkþ1Þ ðxÞ strictly interlaces (resp., Theorem 6. For any n 2 N; T ðmÞ n ðxÞ is real-rooted for m 2 fkþ1 : k 2 Ng # ð0; 1Þ. For n P 2; T n k k k ðkþ1 Þ ðkþ1 Þ ðkÞ ðkþ1 Þ ðkÞ ðxÞ if deg T~ nþ1 ðxÞ ¼ deg T~ nkþ1 ðxÞ þ 1 (resp., deg T~ nþ1 ðxÞ ¼ deg T~ nkþ1 ðxÞ). strictly alternates left of) T~ nþ1
ðkÞ Proof. We first note that all T~ nkþ1 ðxÞ have positive coefficients. k k ð Þ ðkþ1 Þ ðxÞ readily follows from arguments similar to those in the proof of Theorem 3. The case of T~ nkþ1 ðxÞ interlacing T~ nþ1
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ðkÞ In case n þ 1 1ðmod k þ 1Þ, we let xn;1 < xn;2 < < xn;n1bn1c < 0 be the simple zeros of T~ nkþ1 ðxÞ. Define also kþ1 xn;0 :¼ 1. For i ¼ 1; 2; . . . ; n 1 bn1 c, kþ1 k ðkþ1 Þ T~ nþ1 ðxn;i Þ ¼
ð Þ T~ nkþ1 k
0
ðxn;i Þ
k k n1 n1 ð Þ ðkþ1 Þ ðkþ1 Þ so that sgnT~ nþ1 ðxn;i Þ ¼ ð1Þn1bkþ1ci . Since deg T~ nþ1 ðxÞ ¼ n 1 bn1 c, we have sgnT~ nþ1 ðxn;0 Þ ¼ ð1Þn1bkþ1c . Thus, for kþ1 k k ðkþ1Þ ðkþ1 Þ i ¼ 1; 2; . . . ; n 1 bn1 c, there exists xnþ1;i 2 ðxn;i1 ; xn;i Þ such that T~ nþ1 ðxnþ1;i Þ ¼ 0. This accounts for all zeros of T~ nþ1 ðxÞ kþ1 and finishes the proof of the alternating case. The desired result now follows from induction. h k kþ1
k It is remarked that when m 2 ð0; 1Þ n fkþ1 : k 2 Ng, the Touchard polynomial T ðmÞ n ðxÞ has non-real roots. This can be k explained heuristically as follows. When m ¼ kþ1 with k 2 N, the concerned recurrence relations (4.3) and (4.4) both have ðmÞ on their right sides polynomial factors with positive coefficients. This guarantees that the resulting T~ ðxÞ has growing posinþ1
tive coefficients, which observe the Newton inequality [9, p. 104], the latter being a necessary condition for real-rootedness. k for any k 2 N, the recurrence relation then has polynomial multiplicative factor with negative constant term. When m – kþ1 ðmÞ The resulting T~ ðxÞ has smaller coefficients, which possibly violate the Newton inequality, implying that it is not realnþ1
rooted. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
E.T. Bell, Exponential polynomials, Ann. Math. 35 (1934) 258–277. F. Brenti, Unimodal, log-concave and Pólya frequency sequences in combinatorics, Mem. Amer. Math. Soc. 81 (1989) 41. Ch.A. Charalambides, Enumerative Combinatorics, Chapman & Hall, CRC, 2001. Ch.A. Charalambides, O.D. Chrysaphinou, Partition polynomials in fluctuation theory, Math. Nachr. 106 (1982) 89–100. O. Chrysaphinou, On Touchard polynomials, Discrete Math. 54 (1985) 143–152. C.-O. Chow, Counting multiderangements by excedances, Sém. Lothar. Combin. 59 (2008). electronic. G. Dattoli, B. Germano, M.R. Martinelli, P.E. Ricci, Touchard like polynomials and generalized Stirling numbers, Appl. Math. Comput. 12 (2012) 6661– 6665. J. Garloff, D.G. Wagner, Hadamard products of stable polynomials are stable, J. Math. Anal. Appl. 202 (1996) 797–809. G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities, Cambridge University Press, Cambridge, 1952. O.V. Kuzmin, O.V. Leonova, Touchard polynomials and their applications, Disc. Math. Appl. 10 (2000) 391–402. W. Lang, On generalizations of the Stirling number triangles, J. Integer Seq. 3 (2000) 00.2.. T. Mansour, Combinatorics of Set Partitions, Discrete Mathematics and its Applications (Boca Raton), CRC Press, Boca Raton, FL, 2012. T. Mansour, M. Schork, The generalized Touchard polynomials revisited, Appl. Math. Comput. 219 (2013) 9978–9991. M. Mihoubi, Bell polynomials and binomial type sequences, Discrete Math. 308 (2008) 2450–2459. M. Mihoubi, M.S. Maamra, Touchard polynomials, partial Bell polynomials and polynomials of binomial type, J. Integer Seq. 14 (2011) 11.3.. N. Privault, Generalized Bell polynomials and the combinatorics of Poisson central moments, Electron. J. Combin. 18 (2011) 54. J. Touchard, Propriétés arithmétiques de certains nombres recurrents, Ann. Soc. Sci. Bruxelles 53A (1933) 21–31. J. Touchard, Sur les cycles des substitutions, Acta Math. 70 (1939) 243–279.
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
On the real-rootedness of generalized Touchard polynomials Chak-On Chow a, Toufik Mansour b,⇑ a b
Division of Science and Technology, BNU-HKBU United International College, Zhuhai 519085, PR China Department of Mathematics, University of Haifa, 3498838 Haifa, Israel
a r t i c l e
i n f o
a b s t r a c t
Keywords: Generalized Touchard polynomials Real-rootedness Interlacing Alternating
We consider the real-rootedness of generalized Touchard polynomials recently revisited by Mansour and Schork (2013). Towards this end, we first describe the normal form of the generalized Touchard polynomials, by which recurrence relations for the polynomial part are derived. By using the recurrence relations, we prove the real-rootedness of the generaln o k :k2N . ized Touchard polynomials for the parameter m 2 ½1; 1Þ [ kþ1 Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Jacques Touchard [17,18] introduced in 1933 the Touchard polynomials T n ðxÞ’s, in the study of permutations with cycles satisfying certain conditions, as extensions of the partial Bell polynomials [1]. The Touchard polynomials T n ðxÞ’s (also known as the exponential polynomial or Bell polynomial) can either be defined by the exponential generating function P P P j n tn xðet 1Þ or by the Stirling transform T n ðxÞ ¼ nj¼0 Sn;j xj , where Sn;j ¼ 1j! j‘¼0 ð1Þj‘ ‘ denotes the Stirling nP0 T n ðxÞ n! ¼ e ‘ number of the second kind, which counts the number of partitions of an n-set into j non-empty blocks (see [12]). It is well known that the moments of the Poisson distribution are intimately related to the combinatorics of Stirling numbers of the second kind and Bell numbers. More precisely, if the random variable X PoissonðkÞ, then for n ¼ 1; 2; . . ., the Touchard polynomial T n ðxÞ satisfies the relation
T n ðkÞ ¼ E½X n ; i.e., T n ðkÞ is the nth moment of X. Moreover, several properties of Touchard polynomials are studied in [3,5,10,18], where some of them are applied to problems in random walks. Touchard polynomials have been extended in many contexts, see [4,5,10,14–16] for instance. Of relevance to the present work is the higher order extension of Touchard polynomials introduced by Dattoli et al. [7], defined for m 2 Z; n 2 N by n
T nðmÞ ðxÞ :¼ ex ðxm @ x Þ ex ;
ð1:1Þ
which reduce when m ¼ 1 to the classical Touchard polynomials T n ðxÞ mentioned above. Dattoli et al. [7] discussed several properties of these polynomials, including the recurrence (1.2) below for m 2 N (also, see [11]). Mansour and Schork [13] ðmÞ recently revisited T ðmÞ n ðxÞ by extending m to arbitrary real number and showed various properties of T n ðxÞ’s, including the following recurrence relation
⇑ Corresponding author. E-mail addresses: [email protected] (C.-O. Chow), [email protected] (T. Mansour). http://dx.doi.org/10.1016/j.amc.2014.12.108 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
C.-O. Chow, T. Mansour / Applied Mathematics and Computation 254 (2015) 204–209 ðmÞ
ðxm þ xm @ x ÞT ðmÞ n ðxÞ ¼ T nþ1 ðxÞ
205
ð1:2Þ
ð12Þ
as well as an identity relating T n ðxÞ and the nth Hermite polynomial Hn ðxÞ. See (4.1) in Section 4. Since Hn ðxÞ possesses the ð1Þ remarkable property of being real-rooted, this suggests that T n2 ðxÞ enjoys this remarkable property also. A crucial property ð1Þ ð1Þ ð1Þ of T n2 ðxÞ not explored by Mansour and Schork is the representation T n2 ðxÞ ¼ xan T~ n2 ðxÞ, where an 2 0; 12 and ð1Þ ð1Þ ð1Þ T~ n2 2 Qþ ½x. Since the real-rootedness of T n2 ðxÞ implies that of T~ n2 ðxÞ, an application of the Aissen–Schoenberg–Whitney P ð1Þ theorem [2, Theorem 2.2.4] to T~ n2 ðxÞ ¼ di¼0 ai xi then yields the total positivity consequence that any minor of the infinite ð1Þ matrix M ¼ ðM ij Þi;j2N defined by M ij ¼ aji for all i; j 2 N (where ak ¼ 0 if k < 0 or k > d ¼ deg T~ n2 ðxÞ) is non-negative. Because of these, it is of considerable interest to study more generally the real-rootedness of T ðmÞ n ðxÞ for m 2 R. The organization of this paper is as follows. In the next section, we consider the positive integral m case. In Section 3, we consider the negative integral m case. In Section 4, we study how the positive integral m case extends to give the positive real m case. 2. The positive integer case We study in the present section the real-rootedness of T ðmÞ n ðxÞ for m 2 N. Let m 2 N. One can compute T ðmÞ n ðxÞ either directly by (1.1), or by the recurrence (1.2). The first few members are listed as follows: ðmÞ
T 1 ðxÞ ¼ xm ; ðmÞ
T 2 ðxÞ ¼ x2m1 ðx þ mÞ; ðmÞ
T 3 ðxÞ ¼ x3m2 ½x2 þ 3mx þ mð2m 1Þ; ðmÞ
T 4 ðxÞ ¼ x4m3 ½x3 þ 6mx2 þ mð11m 4Þx þ mð2m 1Þð3m 2Þ; ðmÞ
T 5 ðxÞ ¼ x5m4 ½x4 þ 10mx3 þ 5mð7m 2Þx2 þ 5mð2m 1Þð5m 2Þx þ mð2m 1Þð3m 2Þð4m 3Þ: We see that when m 2 N, all coefficients of T ðmÞ n ðxÞ are positive integral. ðmÞ
Proposition 1. For any positive integers m and k; T k ðxÞ is a monic polynomial of degree km with non-negative integer coefficients.
Proof. We fix the positive integer m and proceed by induction on k. Since ðmÞ
T 1 ðxÞ ¼ ex ðxm @ x Þex ¼ xm ; the theorem clearly holds when k ¼ 1. Assume that the result holds up to k. By the recurrence relation ðmÞ
ðmÞ
ðmÞ
ðmÞ 0
T kþ1 ðxÞ ¼ ðxm þ xm @ x ÞT k ðxÞ ¼ xm T k ðxÞ þ xm ðT k Þ ðxÞ; which is a monic polynomial of degree m þ km ¼ ðk þ 1Þm.
h
It is evident from the above list that
T nðmÞ ðxÞ ¼ xnmnþ1 T~ ðmÞ n ðxÞ
ð2:1Þ
ðmÞ ðmÞ for some monic polynomial T~ n 2 N½x with T~ n ð0Þ > 0. We shall show that a sufficient condition for the coefficients of ðmÞ ~ T n ðxÞ to be positive integral for all n is that m 2 N.
ðmÞ Proposition 2. For n 2 N, the Touchard polynomial T~ n ðxÞ is monic, and satisfies the recurrence relation
0 ðmÞ ðxÞ: T~ nþ1 ðxÞ ¼ ðx þ nðm 1Þ þ 1ÞT~ nðmÞ ðxÞ þ x T~ ðmÞ n If m 2 N, then
ðmÞ T~ n
ð2:2Þ
2 N½x.
Proof. Substituting (2.1) into (1.2), we have ðmÞ
0
~ ðmÞ xðnþ1Þmn T~ nþ1 ðxÞ ¼ ðxm þ xm @ x Þxnmnþ1 T~ nðmÞ ðxÞ ¼ xðnþ1Þmn ½ðx þ nm n þ 1ÞT~ ðmÞ n ðxÞ þ xðT n Þ ðxÞ: Cancelling xðnþ1Þmn from both sides, (2.2) follows. The multiplicative factor x þ nm n þ 1 has positive coefficients () m > 1 1n. Thus, for m 2 N, the latter condition holds and both factors on the right side of (2.2) have positive integral ðmÞ coefficients; T~ n 2 N½x then follows by induction. h
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We also need some notations from [8] concerning the relative positions of real zeros of polynomials. A real polynomial f ðxÞ is standard if its leading coefficient is positive; it is simply real-rooted if all its zeros are real and simple. Let f ðxÞ and gðxÞ be simply real-rooted standard polynomials of degrees n and m, respectively, and let h1 < h2 < < hn and n1 < n2 < < nm be their zeros. Then gðxÞ is said to interlace f ðxÞ if n ¼ m þ 1 and
h1 6 n1 6 h2 6 n2 6 6 hn1 6 nn1 6 hn ; gðxÞ is said to alternate left of f ðxÞ if n ¼ m and
n1 6 h1 6 n2 6 h2 6 6 nn 6 hn : In case all the above inequalities are strict, we say that gðxÞ strictly interlaces (resp., strictly alternates left of) f ðxÞ. It is easy to see that if x 2 ðhi ; hiþ1 Þ, then sgnf ðxÞ ¼ ð1Þni , where i ¼ 0; 1; . . . ; n and h0 :¼ 1. This fact will be exploited when proving Theorems 3 and 6. Theorem 3. For any positive integers n and m; T ðmÞ n ðxÞ has the trivial zero x ¼ 0 of multiplicity nm n þ 1, and n 1 simple ðmÞ ðmÞ negative zeros. Moreover, T~ n ðxÞ strictly interlaces T~ nþ1 ðxÞ for n P 2. Proof. We fix the positive integer m and proceed by induction on n, the case n ¼ 1 being trivial by virtue of Proposition 1. ðmÞ Since T~ 2 ðxÞ ¼ 0 () x ¼ m, it follows that ðmÞ T~ 3 ðmÞ ¼ ðmÞ2 þ 3mðmÞ þ mð2m 1Þ ¼ m < 0: ðmÞ ðmÞ Since the graph of T~ 3 ðxÞ is a parabola opening upward, T~ 3 ðmÞ < 0 implies that m must lie in between the zeros of ðmÞ ðmÞ i.e., T~ ðxÞ strictly interlaces T~ ðxÞ. Assume that the result holds for n P 3. Let xn;1 < xn;2 < < xn;n1 < 0 be the
ðmÞ T~ 3 ðxÞ,
2
3
simple zeros of T ðmÞ n ðxÞ. Define also xn;0 :¼ 1 and xn;n :¼ 0. By the recurrence relation (2.2), we have that 0 ðmÞ T~ nþ1 ðxÞ ¼ ðx þ nm n þ 1ÞT~ nðmÞ ðxÞ þ xðT~ ðmÞ n Þ ðxÞ:
Upon setting x ¼ xn;j ; j ¼ 1; 2; . . . ; n 1,
0 ðmÞ T~ nþ1 ðxn;j Þ ¼ xn;j T~ nðmÞ ðxn;j Þ so that ðmÞ sgnT~ nþ1 ðxn;j Þ ¼ ð1Þnj :
ðÞ
ðmÞ ðmÞ Since deg T~ nþ1 ðxÞ ¼ n; sgnT~ nþ1 ðxn;0 Þ ¼ ð1Þn . Since
ðmÞ T~ nþ1 ðxn;n Þ ¼ ðnm n þ 1ÞT~ ðmÞ n ðxn;n Þ > 0; ðmÞ
sgnT~ nþ1 ðxn;n Þ ¼ 1. Thus, (⁄) actually holds for j ¼ 0; 1; . . . ; n. So, for j ¼ 1; 2; . . . ; n, there exists xnþ1;j 2 ðxn;j1 ; xn;j Þ such that ðmÞ ðmÞ ðmÞ ðmÞ T~ nþ1 ðxnþ1;j Þ ¼ 0. These account for those n simple negative zeros of T nþ1 ðxÞ. Since T nþ1 ðxÞ ¼ xðnþ1Þmn T~ nþ1 ðxÞ; x ¼ 0 is a zero ðmÞ
of T nþ1 ðxÞ of multiplicity ðn þ 1Þm n. This finishes the induction and the proof of the theorem.
h
The same idea of proof has been employed by Chow [6] to prove the real-rootedness of the generating function dn ðxÞ of multiderangements by excedances. 3. The negative integer case When m is negative, we shall write T nðmÞ ðxÞ with m 2 N. Mansour and Schork [13] showed that the following recurrence relation ðmÞ
T nþ1 ðxÞ ¼ ðxm þ xm @ x ÞT ðmÞ ðxÞ n T nðmÞ ðxÞ.
holds for A comparison of (1.2) and (3.1) shows that the latter is the former with m in place of m. Let us first inspect the first few members of T nðmÞ ðxÞ, as follows: ðmÞ
T1
ðxÞ ¼ xm ;
ðmÞ T 2 ðxÞ ðmÞ T 3 ðxÞ ðmÞ T 4 ðxÞ ðmÞ T 5 ðxÞ
¼ x2m ð1 mx1 Þ; ¼ x3m ½1 3mx1 þ mð2m þ 1Þx2 ; ¼ x4m ½1 6mx1 þ mð11m þ 4Þx2 mð2m þ 1Þð3m þ 2Þx3 ; ¼ x5m ½1 10mx1 þ 5mð7m þ 2Þx2 5mð2m þ 1Þð5m þ 2Þx3 þ mð2m þ 1Þð3m þ 2Þð4m þ 3Þx4 :
ð3:1Þ
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The above listed members suggest that
T ðmÞ ðxÞ ¼ xnm f n ðx1 Þ; n
ð3:2Þ
where f n 2 Z½x satisfies f n ð0Þ ¼ 1; deg f n ðxÞ ¼ n 1, and the coefficients of f n ðxÞ alternate in signs. Thus, the polynomial f n ðxÞ has all its coefficients positive integral. Proposition 4. For m; n 2 N,
T ðmÞ ðxÞ ¼ xnm f n ðx1 Þ; n where f n 2 Z½x is of degree n 1 and f n ð0Þ ¼ 1. Moreover; g n 2 N½x, where g n ðxÞ :¼ f n ðxÞ. Proof. From the recurrence relation (3.1) and (3.2), we have
xðnþ1Þm f nþ1 ðx1 Þ ¼ ðxm þ xm @ x Þxnm f n ðx1 Þ:
ð3:3Þ
Let y ¼ x1 and g n ðyÞ ¼ f n ðyÞ. Then @ x ¼ dy @ ¼ y2 @ y so that (3.3) becomes dx y
ðyÞðnþ1Þm g nþ1 ðyÞ ¼ ððyÞm þ ðyÞm y2 @ y ÞðyÞnm g n ðyÞ ¼ ðyÞðnþ1Þm g n ðyÞ þ ðyÞm y2 ½ð1Þnm nmynm1 g n ðyÞ þ ðyÞnm g 0n ðyÞ ¼ ðyÞðnþ1Þm ½ð1 þ nmyÞg n ðyÞ þ y2 g 0n ðyÞ: Cancelling ðyÞðnþ1Þm from both sides, we obtain the recurrence relation for g n ðyÞ:
g nþ1 ðyÞ ¼ ð1 þ nmyÞg n ðyÞ þ y2 g 0n ðyÞ: Since g 1 ðyÞ 1, it follows from induction that g n ðyÞ has positive integral coefficients only and g n ð0Þ ¼ 1.
h
With the factorization of T nðmÞ ðxÞ as in Proposition 4, one is tempted to believe that T nðmÞ ðxÞ enjoys a similar real-rootedðmÞ
ðxÞ, it is readily seen that T 2 ness. Unfortunately, this is not the case. From the above list of T ðmÞ n root x ¼ m for any m P 1. But
ð1Þ T 3 ðxÞ
5
ðxÞ ¼ 0 has the only real
2
¼ x ðx 3x þ 3Þ ¼ 0 has non-real roots. Computer evidence further reveals that
ðxÞ ¼ 0 has only real roots, which do not exhibit the usual interlacing property with respect to those when m P 3n; T ðmÞ n ðmÞ
of T nþ1 ðxÞ ¼ 0. Because of the latter observation, we shall not dwell on this case any further. 4. The positive real case With the real-rootedness of T ðmÞ n ðxÞ established in the affirmative in the positive integral m case as in Section 2, we would like to see whether the same is true when m ¼ a 2 Rþ . A closer examination of the proof of Theorem 3 reveals that if the coefficients of T ðmÞ n ðxÞ are positive, not necessarily integral, then the real-rootedness of T ðmÞ n ðxÞ for m P 1 continues to hold, should m be integral or not. It remains to consider the remaining m 2 ð0; 1Þ case, thus completing our discussion of the positive m case. Mansour and Schork [13, Theorem 5.3] showed that
ð1Þ T n2 ðxÞ ¼
n pffiffiffi
i Hn i x ; 2
where Hn ðxÞ is the nth Hermite polynomial defined by the generating function identity: 2
e2tzt ¼
X
Hn ðzÞ
nP0
tn : n!
An alternative way to define the Hermite polynomials is by the classical Rodrigues formula:
Hn ðxÞ ¼ ð1Þn ex
2
n d 2 ex : dx
It is well known that Hn ðxÞ is real-rooted. From (4.1), we see that
ð1Þ T n2 ðx2 Þ ¼
n qffiffiffiffiffiffiffiffiffi i Hn i x2 ¼ 0 2
for any root x 2 R of Hn ðxÞ ¼ 0. It is this particular case which motivates our study of the m 2 ð0; 1Þ case. ðkÞ Let us first inspect the first few members of T nkþ1 ðxÞ with k ¼ 3, as follows:
ð4:1Þ
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ð 3Þ T 14 ðxÞ ¼ x3=4 ; x2=4 ð 3Þ ð4x þ 3Þ; T 24 ðxÞ ¼ 4 x1=4 ð 3Þ T 34 ðxÞ ¼ ð8x2 þ 18x þ 3Þ; 8 1 ð 3Þ T 44 ðxÞ ¼ ð32x3 þ 144x2 þ 102x þ 3Þ; 32 x3=4 ð 3Þ T 54 ðxÞ ¼ ð32x3 þ 240x2 þ 390x þ 105Þ; 32 x2=4 ð 3Þ ð128x4 þ 1440x3 þ 4200x2 þ 3150x þ 315Þ; T 64 ðxÞ ¼ 128 x1=4 ð 3Þ ð256x5 þ 4032x4 þ 18480x3 þ 27300x2 þ 10080x þ 315Þ; T 74 ðxÞ ¼ 256 1024x6 þ 21504x5 þ 142464x4 þ 349440x3 þ 286020x2 þ 51660x þ 315 ð 3Þ : T 84 ðxÞ ¼ 1024 We see that the fractional exponent of x decreases by steps of 14 and the degree of the polynomial part increases by 1. Once the fractional exponent becomes 0, the fractional exponent of the next member starts from 34 with the degree of polynomial part the same as that of the zero exponent member. The pattern repeats itself. Proposition 5. For k; n 2 N,
ðkÞ ðkÞ T nkþ1 ðxÞ ¼ xðkln Þ=ðkþ1Þ T~ nkþ1 ðxÞ;
ð4:2Þ
ðkÞ where n 1 ln ðmod k þ 1Þ with 0 6 ln 6 k, and T~ nkþ1 2 Qþ ½x is of degree n 1 bn1 c satisfying kþ1
k k 0 k k ln ~ ðkþ1 ðkþ1 Þ Þ ð Þ T n ðxÞ þ x T~ nkþ1 ðxÞ T~ nþ1 ðxÞ ¼ x þ kþ1
ð4:3Þ
when n þ 1X1ðmod k þ 1Þ, and
k 0 k ðkþ1 Þ ðkÞ ð Þ T~ nþ1 ðxÞ ¼ T~ nkþ1 ðxÞ þ T~ nkþ1 ðxÞ
ð4:4Þ
otherwise. ðkÞ ðkÞ Proof. It is evident that (4.2) describes the form of T nkþ1 ðxÞ. It remains to show that T~ nkþ1 ðxÞ satisfies the recurrence relations (4.3) and (4.4). For n þ 1X1ðmod k þ 1Þ; k þ 1jn so that n lnþ1 ðmod k þ 1Þ with 0 < lnþ1 6 k. It follows that n 1 ln ¼ lnþ1 1ðmod k þ 1Þ, where 0 6 ln < k. k Setting m ¼ kþ1 and the representation (4.2) into (1.2), we have that
k 0 k k k ln ðkln Þ=ðkþ1Þ1 ~ ðkþ1 ðkþ1 Þ ðkÞ Þ ð Þ T n ðxÞ þ xðkln Þ=ðkþ1Þ T~ nkþ1 ðxÞ x T nþ1 ðxÞ ¼ xk=ðkþ1Þ xðkln Þ=ðkþ1Þ T~ nkþ1 ðxÞ þ kþ1 k k 0 k ln ~ ðkþ1 Þ ð Þ T n ðxÞ þ x T~ nkþ1 ðxÞ ¼ xðkln 1Þ=ðkþ1Þ x þ kþ1
Comparing this equality with (4.2), (4.3) follows. When n þ 1 1ðmod k þ 1Þ; k þ 1jn so that n 1 k ¼ ln ðmod k þ 1Þ and (4.3) becomes (4.4). Note that all multiplicative factors on the right side of (4.3) and (4.4) have non-negative rational coefficients. Since ðkÞ ðkÞ T 1kþ1 ðxÞ ¼ xk=ðkþ1Þ , it follows by induction that T~ nkþ1 ðxÞ has positive rational coefficients only. k k k ðkþ1Þ ð Þ ðkþ1 Þ ðkÞ ðxÞ ¼ deg T~ nkþ1 ðxÞ þ 1. By (4.4), deg T~ nþ1 ðxÞ ¼ deg T~ nkþ1 ðxÞ. This non-increment of degree occurs By (4.3), deg T~ nþ1 ðkÞ c. h whenever n 1 is a multiple of k þ 1. Consequently, we have deg T~ nkþ1 ðxÞ ¼ n 1 bn1 kþ1 k k ~ ðkþ1Þ ðxÞ strictly interlaces (resp., Theorem 6. For any n 2 N; T ðmÞ n ðxÞ is real-rooted for m 2 fkþ1 : k 2 Ng # ð0; 1Þ. For n P 2; T n k k k ðkþ1 Þ ðkþ1 Þ ðkÞ ðkþ1 Þ ðkÞ ðxÞ if deg T~ nþ1 ðxÞ ¼ deg T~ nkþ1 ðxÞ þ 1 (resp., deg T~ nþ1 ðxÞ ¼ deg T~ nkþ1 ðxÞ). strictly alternates left of) T~ nþ1
ðkÞ Proof. We first note that all T~ nkþ1 ðxÞ have positive coefficients. k k ð Þ ðkþ1 Þ ðxÞ readily follows from arguments similar to those in the proof of Theorem 3. The case of T~ nkþ1 ðxÞ interlacing T~ nþ1
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ðkÞ In case n þ 1 1ðmod k þ 1Þ, we let xn;1 < xn;2 < < xn;n1bn1c < 0 be the simple zeros of T~ nkþ1 ðxÞ. Define also kþ1 xn;0 :¼ 1. For i ¼ 1; 2; . . . ; n 1 bn1 c, kþ1 k ðkþ1 Þ T~ nþ1 ðxn;i Þ ¼
ð Þ T~ nkþ1 k
0
ðxn;i Þ
k k n1 n1 ð Þ ðkþ1 Þ ðkþ1 Þ so that sgnT~ nþ1 ðxn;i Þ ¼ ð1Þn1bkþ1ci . Since deg T~ nþ1 ðxÞ ¼ n 1 bn1 c, we have sgnT~ nþ1 ðxn;0 Þ ¼ ð1Þn1bkþ1c . Thus, for kþ1 k k ðkþ1Þ ðkþ1 Þ i ¼ 1; 2; . . . ; n 1 bn1 c, there exists xnþ1;i 2 ðxn;i1 ; xn;i Þ such that T~ nþ1 ðxnþ1;i Þ ¼ 0. This accounts for all zeros of T~ nþ1 ðxÞ kþ1 and finishes the proof of the alternating case. The desired result now follows from induction. h k kþ1
k It is remarked that when m 2 ð0; 1Þ n fkþ1 : k 2 Ng, the Touchard polynomial T ðmÞ n ðxÞ has non-real roots. This can be k explained heuristically as follows. When m ¼ kþ1 with k 2 N, the concerned recurrence relations (4.3) and (4.4) both have ðmÞ on their right sides polynomial factors with positive coefficients. This guarantees that the resulting T~ ðxÞ has growing posinþ1
tive coefficients, which observe the Newton inequality [9, p. 104], the latter being a necessary condition for real-rootedness. k for any k 2 N, the recurrence relation then has polynomial multiplicative factor with negative constant term. When m – kþ1 ðmÞ The resulting T~ ðxÞ has smaller coefficients, which possibly violate the Newton inequality, implying that it is not realnþ1
rooted. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
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