Explicit bounds for multidimensional linear recurrences with restricted coefficients

J. Math. Anal. Appl. 322 (2006) 1159–1167 www.elsevier.com/locate/jmaa

Explicit bounds for multidimensional linear recurrences with restricted coefficients Kenneth S. Berenhaut ∗,1 , John D. Foley Wake Forest University, Department of Mathematics, Winston-Salem, NC 27109, USA Received 19 September 2005 Available online 9 November 2005 Submitted by William F. Ames

Abstract This note employs path counting techniques to extend recent results on bounds for odd order linear recurrences to higher dimensions. The results imply optimal zero-free polydisks for multivariable power series with 0, 1 coefficients. Among the applications is a result that states that the optimal zero-free polydisk has radius approximately 1/(v + 1), for large dimensions v. © 2005 Elsevier Inc. All rights reserved. Keywords: Explicit bounds; Multidimensional recurrences; Multivariable generating functions; Power series; Restricted coefficients; Growth rates; Nonconstant coefficients; Partial order

1. Introduction This paper studies multidimensional linear recurrences with coefficients restricted to the interval [−1, 0]. In particular, suppose that for n = (n1 , n2 , . . . , nv ) ∈ (Z+ )v , we have  αn,m bm (1) bn = m
under some ordering on (Z+ )v , some distance function d and some double sequence {αi,j } satisfying * Corresponding author.

E-mail addresses: [email protected] (K.S. Berenhaut), [email protected] (J.D. Foley). 1 The author acknowledges financial support from a Sterge Faculty Fellowship.

0022-247X/$ – see front matter © 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2005.09.055

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αi,j ∈ [−1, 0],

(2)

for all i, j ∈ (Z+ )v . Specifically, it will be convenient to consider m < n, if and only if mi  ni for  all i and m = n. Also we shall take d to be the taxicab metric (or 1-metric), i.e., d(n, m) = 1iv |ni − mi |. Note that, more generally, one might consider recurrences on any partially ordered set P . In that case the Möbius function of P , i.e., the inverse of its Zeta function, would be an example of a recursive function in the incidence algebra I(P ) with negative coefficients (cf. [18,19,28,29]). In fact, the inverse of any function f ∈ I(P ), with f (x, x) = 1 for all x, that takes values in [0, 1], would have coefficients as in (2). We are interested here in the value of {Un (k)} where, for fixed k,   def Un (k) = max |bn |: {bi }, {αi,j } satisfy (1) and (2) .

(3)

The following was proven for the case v = 1 in [5]. Theorem 1. Suppose v = 1 and k  1 is odd or infinite, and define {Vi } by V(1) = 1, and for n = (n1 ) = (1), Vn =

[k/2] 

(4)

V(n1 −2i−1) .

i=0

Then, Un (k) = Vn for all n. In [5], Theorem 1 was employed to extend a theorem of Graham and Sloane [14] on inverses of (0, 1) matrices. The result to be proven here is the following extension of Theorem 1. Theorem 2. Suppose v  1 and k  1 is odd or infinite, and define {Vi (k)} by V1 = 1 and Vi (k) =



Vj (k).

(5)

j
Then, Ui (k) = Vi (k) for all i. Note than for k even, the corresponding result in Theorem 2 does not hold (see the counterexamples in [5] for the case v = 1). Recurrences satisfying (1) and (2) are closely tied to power series with restricted coefficients (cf. [3,8,9] and the references therein). Implications of Theorem 2 for multivariable power series and polynomials will be discussed in Section 4, below. In particular, we extend a result obtained simultaneously by Flatto, Lagarias, and Poonen [13] and Solomyak [30], for v = 1, to any finite dimension v (see Corollary 4 and Theorem 5, below). The remainder of the paper proceeds as follows. Section 2 contains some preliminary results and notation. In Section 3, we prove Theorem 2, while Section 4 includes discussion of applications to power series.

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2. Preliminary results and notation Suppose {bi } and {αi,j } satisfy (1) and (2) for some v, k  1, with k odd or infinite. Now, Let P = {i: bi  0} and N = {i: bi < 0} partition the sign configuration of {bi }i∈(Z+ )v . Define {Bi } recursively from N and P via B1 = 1 and for i > 1 set ⎧  ⎪ (−1)bj , i ∈ P, ⎪ ⎪ ⎪ ⎪ j
A simple induction with (6) will show that Bi and bi have the same sign for all i ∈ (Z+ )v . The following lemma, similar to its counterparts in [8,9], serves to reduce the problem of bounding |bn | for n ∈ (Z+ )v to a comparison of the at most 2|Z| possible distinct “sign configurations” on Z = {j : j  i}. Lemma 1. We have |bi |  |Bi |

for all i ∈ (Z+ )v .

(7)

Proof. The proof, which follows by induction, is very similar to that for the one-dimensional case (see [8,9]). 2 Following the discretization provided by Lemma 1, we now give a path counting equivalence which proves to be particularly convenient for dealing with the problem at hand, in a combinatorial manner. Specifically, define the sign function, S, on (Z+ )v , via

0, i ∈ P, S(i) = (8) 1, i ∈ N . Since {Bi } be can thought of solely as a function of S, we will sometimes denote Bi via Bi (S). We will consider paths π = p1 , p 2 , . . . , pt  in the sign configuration space ((Z+ )v , S) satisfying p i+1 > p i and S(p i+1 ) = S(p i ) for 1  i  t − 1. For a given set T ⊂ (Z+ )v , sign function S defined on T and initial and ending points m and y, respectively define the T -path-set, ΠT (m, y, S) between m and y, via  ΠT (m, y, S) = π: p 1 = m, p t = y, p i ∈ T for 1  i  t,  (9) and d(pi , p i+1 )  k for 1  i  t − 1 . When T = {i: i < n}, we will use the notation Tn to denote the set T and Πn (S) in place of ΠTn (1, n, S), for simplicity. Example 1. Figure 1 displays the set T(4,3) along with possible values for S. Here we use “+” in the ith position, if S(i) = 0 (i.e., i ∈ P) and “−” in the ith position, if S(i) = 1 (i.e., i ∈ N ). In the case depicted in Fig. 1, if k = 3 then (1, 1), (1, 3), (2, 3), (4, 3), (1, 1), (2, 1), / Π(4,3) (S), (2, 2), (3, 2), (4, 2), (4, 3) would be elements of Π(4,3) (S), while (1, 1), (4, 3) ∈ since d((1, 1), (4, 3)) = 5 > 3. The proof of the following is similar to that of [6, Lemma 2] (which deals with the case v = 1, and will be omitted.

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i2

3 2 1

−+ + − −+ −+ +− + − 1

2

3

i2

4

6 5 4 3 2 1

− + − +−+−+ −+−+− +−+−+ 1

2

Fig. 1.

3

4

5

i1

i1 Fig. 2.

Theorem 3. For all n ∈ (Z+ )v , and any S, we have     Bn (S) = Πn (S). Proof. Similar to the proof in the one-dimensional case (see [6]).

(10) 2

For a given vector n = (n1 , n2 , . . . , nv ) and an extension constant c, define Tn,c = Tn ∪ {i: i = (n1 , n2 , . . . , nv + j ), 0  j  c}. In particular, for c = 0, we have Tn,0 = Tn . In addition, we will use the notation Πn,c (S) for ΠTn,c (1, (n1 , n2 , . . . , nv + c), S). Example 2. Figure 2 depicts the set T(5,3),3 with S(i) taken to be the parity of d(1, i) for each i. We now turn to a proof of Theorem 1. 3. Proof of Theorem 1 The main result of this section is the following. Theorem 4. If k is odd or infinite, then for any n ∈ (Z+ )v and c  0,     maxΠn,c (S) = Πn,c (S  ), S

(11)

where S  (i) is defined to be the parity of d(1, i) for each i. Note that |Πn,0 (S  )| = |Πn (S  )| = Vn . Applying Theorem 3, we have the following corollary, from which Theorem 2 follows directly. Corollary 1. If k is odd or infinite, then for any n ∈ (Z+ )v     maxBn (S) = Bn (S  )

(12)

and consequently   max |bn |: {bi }, {αi,j } satisfy (1) and (2) = Vn .

(13)

S

Proof of Theorem 4. Suppose n = (n1 , n2 , . . . , nv ) ∈ (Z+ )v and c  0 are fixed, and let H = 1iv−1 ni . The key to the proof is a splitting of the set Tn,c into H sets, each of smaller size, for which the optimization can be performed separately. As an example, consider the set T(5,3),3 ⊂ (Z+ )2 (see Example 2). For a given S, the set of paths Π(5,3),3 (S) can be partitioned into H = 5 subsets according to which of the sets

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{(1, 3)}, {(2, 3)}, {(3, 3)}, {(4, 3)} or {(5, 3), (5, 4), (5, 5), (5, 6)} is visited first by a given path. Those paths which first visit (1, 3) can be considered as being composed of two parts, namely, a path through {(1, 1), (1, 2), (1, 3)} and one through {(1, 3), (2, 3), (3, 3), (4, 3), (5, 3), (5, 4), (5, 5), (5, 6)}. The first of these sets can be expressed as T(1,2),1 while the second (once translated, and straightened) can be viewed as being of the same form (relative to the distance function d) as T(8,1) . Similar partitionings are possible for the sets of paths which first visit {(2, 3)}, {(3, 3)}, or {(4, 3)}. The remaining paths, namely those which do not visit the set {(1, 3), (2, 3), (3, 3), (4, 3)}, are simply those through T(5,2),4 . Now, note that for v = 1, T(i),c = T(i+c) , and hence Theorem 1 gives (11) for n = (i) and c  0. Hence, suppose (11) holds for all c  0 and n ∈ (Z+ )v for v < V . If n = (n1 , n2 , . . . , nV −1 , 1) ∈ (Z+ )V , then paths through Tn,c may be viewed as paths through Tn ,c where n = (n1 , n2 , . . . , nV −1 ) ∈ (Z+ )V −1 and hence (11) for such n follows from the v = V − 1 case. Thus, suppose the equation holds for all n = (n1 , n2 , . . . , nV −1 , l) ∈ (Z+ )V with l < L. For given n with nV = L, as in the above example for v = 2, partition the paths through Tn,c into H = 1iV −1 ni sets corresponding to the first visit to a vector m = (m1 , m2 , . . . , mV −1 , L), i.e., consider the H − 1 sets  Pm1 ,m2 ,...,mV −1 (S) = π = p 1 , p 2 , . . . , p t  ∈ Πn,c (S): ∃z such that

 pz = (m1 , m2 , . . . , mV −1 , L) and (p y )V < L for y < z ,

where m = n, and the residual set

  ∗ P (S) = Πn,c (S)

 Pm1 ,m2 ,...,mV −1 (S) .

(14)

(15)

m1 ,m2 ,...,mV −1

Note that the sets in (14) and (15), together, partition Πn,c (S). For counting purposes, the set of paths in Pm1 ,m2 ,...,mV −1 (S) can be split into two parts, namely def

portions in Π(m1 ,m2 ,...,mV −1 ,L−1),1 (S), and those in Π ∗ = ΠT ∗ (m, (n1 , n2 , . . . , nV −1 , L + c), S), where   T ∗ = (q1 , q2 , . . . , qV −1 , L): mi  qi  ni for 1  i  V − 1   ∪ (n1 , n2 , . . . , nV −1 , l): L  l  L + c .

(16)

Since there is a one-to-one correspondence between paths in Π ∗ and paths in Πn−m+1,c (S), applying the induction hypothesis, we have  Pm

1 ,m2 ,...,mV −1

    (S) = Π(m1 ,m2 ,...,mV −1 ,L−1),1 (S)Πn−m+1,c (S)     Π(m1 ,m2 ,...,mV −1 ,L−1),1 (S  )Πn−m+1,c (S  )   = Pm1 ,m2 ,...,mV −1 (S  ).

(17)

For the set P ∗ (S), we have that paths here are specifically those through T(n1 ,n2 ,...,nV −1 ,nV −1),c+1 . Hence by induction we have  ∗   P (S) = ΠT

(n1 ,n2 ,...,nV −1),c+1

     (S)  ΠT(n1 ,n2 ,...,nV −1),c+1 (S  ) = P ∗ (S  ).

Combining (17) and (18) then gives the result.

2

(18)

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4. Applications of Theorem 2 to reciprocals and zero-free regions of power series For a fixed I ⊂ , let FI be the set of I -power series defined by   ∞  i ai z and ai ∈ I for each i  1 . FI = f : f (z) = 1 +

(19)

i=1

Flatto, Lagarias, and Poonen [13] and Solomyak [30] proved independently that if z is a root √ √ of a series in F[0,1] , then |z|  ρ0 , where ρ0 is the golden ratio 2/(1 + 5). As z = −2/(1 + 5) is a root of 1 + z + z3 + z5 + · · ·, this bound is tight over F[0,1] . Note that this result also follows directly from results in [14,19]. By considering coefficients of reciprocals of series in FI , similar zero free regions have been obtained for a variety of intervals I (cf. [4,9]). (v) More generally, for a fixed I ⊂ , let FI be the set of v-variate, I -power series defined by 

 (v) i + v (20) ai z , a0 = 1, and ai ∈ I for each i ∈ (Z ) . FI = f : f (z) = i∈Nv

For some background on analysis with so-called multiple series and power series in several complex variables see, for instance, [24,26,27]. While similar computations would be possible for other values of v, for simplicity, we will restrict attention here to the two-dimensional case (i.e., v = 2). We will also write, for instance, fi,j in place of f(i,j ) .  (2) The coefficients of the multiplicative inverse h of a series f = fi,j x i y j ∈ F[0,1] may be computed by equating coefficients in the expansion      (21) hi,j x i y j fi,j x i y j = 1. This results in h0,0 = 1, and for (n, m) = (0, 0),  hn,m = − fn−i,m−j hi,j .

(22)

in; j m (i,j )=(n,m)

In particular, if fi,j = 0 whenever i +j > k (i.e., f is a polynomial in x and y with total degree less than or equal to k), then the coefficient fn−i,m−j in (22) is zero whenever d((n, m), (i, j )) > k and we have a kth order recurrence as in (1):  (−fn−i,m−j )hi,j . (23) hn,m = (i,j )<(n,m) d((n,m),(i,j ))k

Theorem 2 is, then, directly applicable and leads to the following corollary. (2)

Corollary 2. Suppose that f ∈ F[0,1] has total degree k, with k odd or infinite, and f and h satisfy (21). Then, |h(i,j ) |  V(i+1,j +1) (k) for all (i, j ) ∈ N2 where {Vi (k)} is as in (5).

(24)

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Table 1 Some values of V(i,j ) (∞) i/j 1 2 3 1 1 1 1 2 1 2 4 3 1 4 10 4 2 8 23 5 3 15 50 6 5 28 104 7 8 51 210 8 13 92 414 9 21 164 801 10 34 290 1526

4 2 8 23 60 144 328 718 1524 3156 6404

5 3 15 50 144 378 931 2187 4952 10 886 23 354

6 5 28 104 328 931 2458 6150 14 756 34 230 77 220

7 8 51 210 718 2187 6150 16 296 41 223 100 448 237 293

8 13 92 414 1524 4952 14 756 41 223 109 500 279 251 688 508

9 21 164 801 3156 10 886 34 230 100 448 279 251 743 102 1 906 931

1165

10 34 290 1526 6404 23 354 77 220 237 293 688 508 1 906 931 5 081 378

Table 1 displays the values of V(i,j ) (∞) for i, j  10. It is not difficult to show, from (5), that {V(i,j ) (∞)} satisfies the simpler recurrence V(n1 ,n2 ) (∞) = V(n1 −1,n2 ) (∞) + V(n1 −2,n2 ) (∞) + V(n1 ,n2 −1) (∞) + V(n1 ,n2 −2) (∞) − V(n1 −2,n2 −2) (∞),

(25)

whenever n1 , n2 > 3. Bounds of the sort in Corollary 2 may be useful where generating functions or formal power series are utilized such as in enumerative combinatorics and in the study of stochastic processes (cf. Wilf [31], Feller [12], Kijima [17], Heathcote [15], Kendall [16], Berenhaut and Lund [7] and Berenhaut et al. [3]). For further discussion of multivariable generating functions and related multivariate linear recurrences, cf. [21–23], and the references therein. As in the case of v = 1, results of the sort in Corollary 2 can also provide bounds for the location of the smallest root of associated complex valued power series. Power series with restricted coefficients, in the single variate case, have been studied in the context of determining distributions of zeroes (cf. Flatto et al. [13], Solomyak [30], Beaucoup et al. [1,2], and Pinner [25]). Related problems for polynomials have been considered by Odlyzko and Poonen [20], Yamamoto [32], Borwein and Pinner [11], and Borwein and Erdélyi [10]. As mentioned above, (1) Flatto et al. [13] and Solomyak [30] independently proved that if z is a root of a series in F[0,1] , √ then |z|  r0−1 , where r0 = (1 + 5)/2) ≈ 1.618034. The following related result is an immediate consequence of Corollary 2 and (25) and Abel’s lemma (cf. [24,26,27]). (2)

(2)

Corollary 3 (Zero-free region for elements of F[0,1] ). Suppose f ∈ F[0,1] , and (x1 , x2 ) is inside the bidisk {|xj | < r −1 , j = 1, 2}, where r ≈ 2.69173951 is the largest real root of x 4 − 2x 3 − 2x 2 + 1 = 0. Then h(x1 , x2 ) = L < ∞, and hence f (x1 , x2 ) = 1/L = 0. Note that the series consisting of all odd degree terms, i.e., f (x, y) = 1 + x + y + x 3 + x 2 y + has a zero at (x, y) = (−r −1 , −r −1 ) and the bidisk given in the corollary is optimal. Similar reasoning leads to the following result for general v. xy 2 + y 3 + x 5 + · · ·,

(v) (v) ). Suppose f ∈ F[0,1] , and x = (x1 , x2 , Corollary 4 (Zero-free region for elements of F[0,1] −1 . . . , xk ) is inside the polydisk {|xj | < rv , 1  j  v}, where rv is the largest real root of the polynomial gv , given by

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Table 2 Radii of optimal polydisks

v

v

rv

rv−1

1 2 3 4 5 6 7 8 9

1.618033989 2.691739510 3.791140848 4.828427125 5.854101966 6.872983346 7.887482194 8.898979486 9.908326913

0.6180339887 0.3715069739 0.2637728431 0.2071067812 0.1708203933 0.1454972244 0.1267831705 0.1123724357 0.1009252126

v −1

gv (X) = X 2 − vX 2

v −2

− vX 2

+ v − 1.

(26)

Then, f (x) = 0. Note that gv (1) < 0 and gv is of even degree and hence rv ∈ (1, ∞). As in the case v = 2, the series consisting of all odd degree terms has a zero at x = (−rv−1 , −rv−1 , . . . , −rv−1 ) and hence, the polydisk given in the corollary is optimal. Table 2 gives the (numerically computed) radii of the optimal zero-free polydisks for small v. Note that   v gv (X) = X 2 −3 2v X 2 + (−v2v + v)X + (2v − v2v ) (27) and gv has three real roots, namely, 0 and (upon simplification),     1 1 v2 1 v 1 − v−1 v(v + 4) + v 1− v − x1 (v) = 2 2 2 2 2 and

    1 1 v2 1 v 1 − v−1 v(v + 4) + v . 1− v + x2 (v) = 2 2 2 2 2

(28)

(29) v −1

For all v  1, we have x1 (v) < x2 (v), x2 (v) > 0, gv (v) = −v 2 v (v + 1)2 −2 + v − 1 > 0. Also, for v  2, x2 (v) > v, and   lim x2 (v) − (v + 1) = 0. v→∞

+ v − 1 < 0 and gv (v + 1) = (30)

Combining these facts with 1 < r1 < 2 gives the following result. Theorem 5. Suppose v  1, and let rv−1 be the radius of the optimal zero-free v-dimensional polydisk defined in the statement of Corollary 4. Then, we have v < rv < v + 1 and rv lim = 1. (31) v→∞ v + 1 References [1] F. Beaucoup, P. Borwein, D.W. Boyd, C. Pinner, Power series with restricted coefficients and a root on a given ray, Math. Comput. 222 (1998) 715–736.

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