Approximations of Analytic Functions via Generalized Power Product Expansions

Available online at www.sciencedirect.com

ScienceDirect Journal of Approximation Theory 188 (2014) 19–38 www.elsevier.com/locate/jat

Full length article

Approximations of Analytic Functions via Generalized Power Product Expansions H. Gingold a , Jocelyn Quaintance b,∗ a West Virginia University, Department of Mathematics, Morgantown WV 26506, USA b Rutgers University, Department of Mathematics, Piscataway NJ 08854, USA

Received 2 October 2013; accepted 29 August 2014 Available online 21 September 2014 Communicated by Doron S. Lubinsky

Abstract  n series. For a fixed set of nonzero complex numbers Let f (x) = 1 + ∞ n=1 an x be a formal power ∞ k rk {rk }∞ we convert f (x) into the formal product k=1 (1+gk x ) , namely the Generalized Power Product k=1 Expansion. We provide estimates on the domain of absolute convergence of the infinite product when  n is absolutely convergent. This makes it possible to use the truncated Generalized a x f (x) = 1 + ∞ n n=1 M (1 + gk x k )rk as approximations to the analytic function f (x). The results Power Product Expansions n=1 are made possible by certain intriguing algebraic properties characteristic of the expansions for the case of rk ≥ 1. An asymptotic formula for the gk associated with the majorizing power series is provided. A combinatorial interpretation of the Generalized Power Product Expansion with {rk }∞ k=1 being integers is also given. c 2014 Elsevier Inc. All rights reserved. ⃝

MSC: 41A10; 30E10; 05A17; 11P81 Keywords: Power products; Generalized power products; Power series; Analytic functions; Expansions; Convergence; Partitions; Multi-sets

∗ Corresponding author.

E-mail addresses: [email protected] (H. Gingold), [email protected] (J. Quaintance). http://dx.doi.org/10.1016/j.jat.2014.08.006 c 2014 Elsevier Inc. All rights reserved. 0021-9045/⃝

20

H. Gingold, J. Quaintance / Journal of Approximation Theory 188 (2014) 19–38

1. Introduction ∞ n Consider an infinite power series ∞ 1+ nn=0 an x that when convergent represents the complex valued function f (x) = 1 + n=1 an x . The subject of this paper is the conversion of the  power series with complex coefficients into an infinite product f (x) = 1 + ∞ a xn = n n=1 r 2 r k r k 1 2 (1 + g1 x) (1 + g2 x ) · · · (1 + g k x ) · · · , where {rk } is an arbitrary set of nonzero ∞ k rk is the Generalized Power Product complex numbers. The expression k=1 (1 + gk x ) Expansion, denoted GPPE, and provides a factorization of f (x). Finite truncations of the GPPE,  ∞ M k rk , provide polynomial approximations for f (x). k=1 (1 + gk x ) M=1

Special cases of GPPE appear throughout the literature. The infinite product with elementary factors (1+x k ) is used as a generating function to derive the coefficients q(n) in the power series ∞  k=1

(1 + x k ) = 1 +

∞ 

q(n)x n .

(1.1)

n=1

 k rk It is the special case of ∞ k=1 (1 + gk x ) with gk = 1 and rk = 1. Equally important is the infinite product known to Euler and his successors, ∞  k=1

(1 − x k )−1 = 1 +

∞ 

p(n)x n .

(1.2)

n=1

Each p(n) of a given nonnegative integer into unrestricted parts [2,14]. Eq.  counts the partitions k )rk with g = −1 and r = −1. (1.2) is ∞ (1 + g x k k k k=1 As these two classical examples suggest, the expansion of a general infinite product ∞ k )rk into a power series 1+Σ ∞ a x n generates an infinite sequence of coefficients (1+g x k n k=1 n=1 an that count the number of arrangements in a variety of combinatorial configurations. Various combinatorial interpretations for the discussed in [11]. case of rk = 1k are rk and its companion power series are very The convergence properties of ∞ (1 + g x ) k k=1 important for similar reasons [15,25]. They are crucial in determining the order of growth of the coefficients an = q(n), p(n), as n goes to infinity. See [2] for older results and [12,23,22] for contemporary work. In the twentieth century analysts investigated the convergence of the infinite product ∞ ∞ a x n , a ∈ C. The expression (1 + gk x k ) obtained from the power series f (x) = 1 + Σn=1 n n k=1 ∞ k k=1 (1 + gk x ) is called the Power Product Expansion of f (x). Working independently several mathematicians developed expressions for the coefficients gk in terms of the coefficients {an }∞ n=1 . The earliest systematic approach goes back to Ritt [25]. Significant work was also done by Borofsky, Feld, Hertzog, Ketchum, Kolberg, A. Knopfmacher, Indlekofer, and Warlimont [5–8,16,18,24,20,19,11,9,12,13,17,21]. Most of these works focused primarily on estimates of the radius of convergence of the Power Product Expansion. The typical result being an estimate of the radius of convergence in terms of the radius of convergence found via logarithmic derivative ∞ f ′ (z) n−1 . The sharpest result in this direction was obtained independently in n=1 dn z f (z) = 1

[17,21]. It stated that the Power Product Expansion converges for |z| < [sup |dk | k ]−1 , the supremum being taken over all positive integers k. There are advantages and insights gained by a logarithmic derivative. However, they come with a penalty. We get an indirect expression for the coefficients gk in terms of the coefficients d1 , d2 , . . . , dk rather than a direct expression of gk in terms of the a1 , a2 , . . . , ak . Consequently,

H. Gingold, J. Quaintance / Journal of Approximation Theory 188 (2014) 19–38

21

an estimate for the radius of convergence of the Power Product Expansion is not directly expressed in terms of the order of growth of the coefficients a1 , a2 , . . . , ak . This shortcoming was remedied by H. Gingold and A. Knopfmacher who expressed gk as a polynomial in the variables {a j }kj=1 [12,10]. From the structure of these polynomials an estimate for the radius of convergence of the Power Product Expansion was found in terms of {ak }∞ k=1 . The order of presentation in this work is as follows. In Section 2 we study the expansion of a power series into a GPPE and provide three algebraic representations for the coefficients gk as ∞ polynomials of the {an }∞ n=1 whose coefficients are rational functions of {rn }n=1 . The algebraic results reveal an intriguing property of these expansions. If an ≤ 0 and rn ≥ 1 for all n, then the coefficients gk in the GPPE are non-positive. Moreover, this non-positivity of coefficients is characteristic to a myriad of other related coefficients. In Section 3 we exploit this structure property to determine convergence conditions of the GPPE. We also provide an asymptotic formula for  1 n n n the gk associated with 1 − ∞ n=0 s x , where s = supn≥1 |an | . We end with Section 4 where we provide additional combinatorial interpretations for GPPE in term of partitions of multi-sets. 2. Algebraic formulas for the coefficients of a Generalized Power Product Expansion  n Given a formal power series 1 + ∞ function f with f (0) = 1 and a n=1 an x or an analytic n , we define the Generalized Power Taylor power series representation f (x) = 1 + ∞ a x n=1 n Product Expansion or GPPE of f (x) as f (x) =

∞ 

(1 + gk x k )rk = (1 + g1 x 1 )r1 (1 + g2 x 2 )r2 (1 + g3 x 3 )r3 · · · ,

(2.1)

k=1 ∞ for certain coefficients {gk }∞ k=1 and a set of arbitrary nonzero complex numbers {rk }k=1 . In the context of formal power series we may expand each factor on the right side of Eq. (2.1) via Newton’s Binomial Theorem as follows. ∞   ∞ ∞   ∞   k3 k  k    r3 r2 r1 2 2 k 1 1 · · · . (2.2) 1+ ak x = g2 x g3 x 3 g1 x k1 k2 k1 k=1 k =0 k =0 k =0 1

2

3

By comparing the coefficients of x k in Eq. (2.2) we discover that   r    rl  rl θ k 1 glv11 · · · glvθθ , ak = gk + ··· v 1 v θ 1 l·v=k

(2.3)

l j
where l = [l1 , l2 , . . . , lθ ] and v = [v1 , v2 , . . . , vθ ]. Solving Eq. (2.3) for gk gives us        1  rl 1 rl θ  gk = ··· glv11 · · · glvθθ  . ak − rk v1 vθ l·v=k

(2.4)

l j
Eq. (2.2) of [12] is now a special case of Eq. (2.4) for rk = 1, k = 1, 2, . . . . ∞ ∞ We can develop two other formulas which relate {gn }∞ n=1 in terms of {an }n=1 and {rn }n=1 .  k−1 n k ∞ (−1) (gn x ) , where we say log (1 + gn x n ) exists and is Define log (1 + gn x n ) := k=1 k

22

H. Gingold, J. Quaintance / Journal of Approximation Theory 188 (2014) 19–38

well-defined if the series itself converges. Next define ∞ 

∞ ∞     (−1)k−1 (gn x n )k rn log 1 + gn x n := rn , (2.5) k n=1 n=1 k=1  n by the convention that ∞ n=1 rn log (1 + gn x ) exists and is well-defined if the double sum on the right side of Eq. (2.5) converges. Notice that if the double sum on the right of Eq. (2.5) is  ∞ (−1)k−1 (gn x n )k n n absolutely convergent both ∞ n=1 rn log (1 + gn x ) and log (1 + gn x ) = k=1 k are also absolutely convergent. Using Eq. (2.5) we define

log f (x) :=

∞ 

rn log(1 + gn x n ).

(2.6)

n=1

 n Assume that ∞ n=1 rn log(1 + gn x ) is absolutely convergent. Differentiating both sides of Eq. (2.6) implies that ∞ ∞ ∞    f ′ (x) nrn gn x n−1 n−1 = nr g x (−gn x n )s = n n n f (x) 1 + g x n n=1 s=0 n=1

=

∞  ∞  (−1)s nrn gns+1 x ns+n−1 . s=0 n=1

Let

f ′ (x) f (x)

= d0 +

∞

k=1 dx x

k.

This means we can write the previous line as

∞  ∞ ∞ ∞    (−1)s nrn gns+1 x ns+n−1 = d0 + dk x k = r1 g1 + dk x k . s=0 n=1

k=1

(2.7)

k=1

By comparing the coefficients x k in the two series expansions of Eq. (2.7) we deduce that dk = −



(−1)

k+1 n

k+1

nrn gn n .

(2.8)

n:n|(k+1)

If k + 1 is prime, n = 1 or n = k + 1, and Eq. (2.8) becomes dk = (−1)k r1 g1k+1 + (k + 1)rk+1 gk+1 . If we solve this equation for gk+1 we discover that gk+1 =

dk + (−g1 )k+1r1 , (k + 1)rk+1

k + 1 prime.

(2.9)

Eq. (15) on p. 148 [11] is now a special case of Eq. (2.9). If k + 1 is not prime Eq. (2.8) implies that (k + 1)rk+1 gk+1 = dk +



(−1)

k+1 n

k+1

nrn gn n .

n|(k+1) n̸=k+1

Eq. (2.10) allows us to recursively solve for gk in terms of {g j }k−1 j=1 and dk−1 .

(2.10)

H. Gingold, J. Quaintance / Journal of Approximation Theory 188 (2014) 19–38

23

∞ ∞ There is another way  to define {gn }∞ n=1in terms of {an }n=1 and {rn }n=1 . To begin restate ∞ ∞ n n r n Eq. (2.1) as f (x) = 1 + n=1 A1,n x =  n=1 (1 + gn x ) , where A1,n = an for all positive ∞ n rn n integers n. The key is to recursively define ∞ n= j (1+gn x ) = 1+ n= j A j,n x , substitute this n definition into the restatement of Eq. (2.1), and compare the coefficients of x . These definitions provide the following recursive system of equations. ∞  (1 + gn x n )rn A1,n x n = (1 + g1 x)r1 n=1   n=2 ∞  = (1 + g1 x)r1 1 + A2,n x n

f (x) = 1 +

∞ 

n=2

1+

∞ 

A2,n x n = (1 + g2 x 2 )r2

n=2

∞ 

(1 + gn x n )rn

n=3 

= (1 + g2 x )

2 r2

1+

∞ 

 A3,n x

n

n=3

.. . 1+

∞ 

∞ 

A j,n x n = (1 + g j x j )r j

n= j

(1 + gn x n )rn

n= j+1

 = (1 + g j x )

j rj

1+



∞ 

A j+1,n x

n

n= j+1

.. ..

Newton’s Binomial Theorem allows us to write the previous equation as   ∞ ∞   n n j rj 1+ A j+1,n x 1+ A j,n x = (1 + g j x ) n= j

n= j+1

 = 1+

∞    rj k=1

k

 g kj x jk

1+

∞ 

 A j+1,n x n .

(2.11)

n= j+1

Comparing the coefficients of x N for N = jm + p, where 0 ≤ p ≤ m − 1, provides j different recurrences which we summarize as   N

A j+1,N = A j,N −

j   rj 

k=1

k

g kj A j+1,N − jk .

If N = j this recurrence implies that g j =

A j, j rj

(2.12)

. We may then rewrite Eq. (2.12) as

  N

A j+1,N = A j,N −

j  k  r j  A j, j A j+1,N − jk

k=1

k

r kj

.

(2.13)

24

H. Gingold, J. Quaintance / Journal of Approximation Theory 188 (2014) 19–38

By iterating Eq. (2.13) r times we obtain   N j

A j+1,N = A j,N +  N − j (k

N − jk1 j

  (−1)r



r =1

k2 =1

1 +k2 +···+kr −1 j



k1 =1  )



kr =1

rj k1





···

rj k2



 ···

rj kr



φ,

(2.14)

where φ :=

2 +···+kr Akj,1 +k j

r kj 1 +k2 +···+kr

A j,N − j (k1 +k2 +···+kr ) .

When calculating Eq. (2.12) we used 1+

∞ 

 A j,N x

N= j

N

= (1 + g j x )

j rj

1+

∞ 

 A j+1,N x

N

,

(2.15)

N = j+1

and compared the coefficient of x N . Let us now look at a modified version of Eq. (2.15), namely   ∞ ∞   N N j −r j . (2.16) 1+ A j,N x 1+ A j+1,N x = (1 + g j x ) N = j+1

N= j

Comparing the coefficient of x s in Eq. (2.16) provides surprising insight into   the structure of   k x+k−1 allow us to write A j+1,s . Newton’s Binomial Theorem and the fact that −x = (−1) k k the right side of Eq. (2.16) as     ∞ ∞  ∞    −r j N j k N 1+ A j+1,N x = 1 + (g j x ) 1+ A j,N x k N= j N = j+1 k=1      ∞ ∞   j k N k rj + k − 1 (g j x ) 1+ A j,N x = 1+ (−1) k N= j k=1      k ∞ ∞   A j, j jk k rj + k − 1 N = 1+ (−1) x 1+ A j,N x , k r kj N= j k=1 where the last equality uses the observation that g j =

A j  j,  . rj 1

If we compare the coefficient of x s on both sides of the previous equation we discover that   r j +k−1  k A j+1,s = (−1)k Akj, j A j,N . (2.17) k r j N + jk=s Eq. (2.17) is the key to proving two important theorems about A j+1,s .

25

H. Gingold, J. Quaintance / Journal of Approximation Theory 188 (2014) 19–38

Theorem 2.1. Let j be any positive integer. Define A j,0 = 1 and A j,N = 0 for 1 ≤ N ≤ j − 1. Assume that r j ≥ 1 for all j and that A j,N ≤ 0 for all j ≤ N . Then A j+1,s ≤ 0 whenever j + 1 ≤ s. Proof. Eq. (2.17) is equivalent to  

A j+1,s =

(−1)k

N + jk=s N ̸=0, j

 s

+ (−1) j

−1

r j +k−1 k r kj

r j + sj −2 s j −1

r





r j + sj −1



s j s j

Akj, j A j,N +

s

(−A) j,j j

rj

 s

A j,j j .

s j −1

(2.18)

Set  A :=



(−1)k

N + jk=s N ̸=0, j



r j + sj −1

 Akj, j A j,N , 



s j s j

B :=

r j +k−1 k r kj

s

(−A) j,j j −

rj

r j + sj −2 s j −1

r



s j −1

s

(−A) j,j j .

Eq. (2.18) is A j+1,s = A + B where B corresponds to the two terms provided by N = 0 and N = j. 

We begin by analyzing the structure of A. If r j ≥ 1 the factor C := (r j +k−1)(r j +k−2)···r j k!r kj

r j +(k−1) k r kj



=

is always positive. By hypothesis A j, j ≤ 0 and A j,N ≤ 0. We must analyze

the sign of P := Akj, j A j,N . This is a product with k + 1 factors. If k is even, say k = 2m, P is nonpositive since it is the product of k + 1 = 2m + 1 nonpositive numbers. Thus (−1)k C P = (−1)2m C P = C P is also nonpositive since C is always positive. If k is odd, say k = 2m + 1, P is nonnegative since it is the product of k + 1 = 2m + 2 nonpositive numbers. However (−1)k C P = (−1)2m+1 C P = −C P is nonpositive since C is positive and P is nonnegative. This analysis shows that all the summands in A are nonpositive under given hypothesis. We now analyze the structure of B. First observe that B vanishes unless k is a multiple of s. So assume sj = kˆ where kˆ > 1. Then  B =

ˆ r j +k−1 ˆk



ˆ r kj

 ˆ

 =

r j + kˆ − 1 kˆ

ˆ

(−A)kj, j + (−1)k−1 ˆ r j +k−2 ˆ k−1 ˆ r kj

ˆ r j +k−2 ˆk−1



ˆ r k−1 j



ˆ

Akj, j 

ˆ

ˆ

(−A)kj, j + (−1)k−1

ˆ r j +k−2 ˆ k−1 ˆ r k−1 j

 ˆ

Akj, j

26

H. Gingold, J. Quaintance / Journal of Approximation Theory 188 (2014) 19–38

 ˆ k−1

= (−1)

= (−1)

ˆ

If r j ≥ 1 then

ˆ (r j −1)(k−1) r j kˆ

ˆ r j +k−2 ˆ k−1

ˆ

 (r j − 1)(kˆ − 1) . r j kˆ

(2.19)

ˆ

is positive. By hypothesis A j, j ≤ 0. Thus the sign of Akj, j is (−1)k

ˆ

r k−1 j

 ˆ Akj, j



r k−1 j

ˆ r j +k−2 ˆ k−1

 r j + kˆ − 1 − +1 r j kˆ



r k−1 j



and

ˆ r j +k−2 ˆ k−1

 ˆ Akj, j

ˆ

ˆ k−1

ˆ (−1)k−1



r k−1 j 



ˆ r j +k−2 ˆ k−1

ˆ Akj, j is nonpositive. On the other hand r j ≥ 1, with kˆ > 1, implies that

is positive or zero. Thus the representation of B provided by Eq. (2.19) shows that B

is either zero or negative.



If we use the notation of [10] we may transform Theorem 2.1into a theorem about the structure of the A j+1,s . Define α = ( j1 , j2 , . . . , jn ) to be a vector with n components where each component is a positive integer. Let λ = λ(α) bethe length of α, i.e. λ = n. Let |α| den note the sum of the components, namely |α| = s=1 js . The symbol A j,α represents the expression A j, j1 A j, j2 · · · A j, jn . For example if α = (2, 3, 4, 3), then λ = 4, |α| = 12 and A j,(2,3,4,3) = A j,2 A j,3 A j,4 A j,3 = A j,2 A2j,3 A j,4 . Theorem 2.2. Let j be a positive integer. Then  A j+1,s = (−1)λ(α(l))−1 |c(α(l), j, s)|A j,α(l) l

 = (−1)λ(α(l))+1 |c(α(l), j, s)|A j,α(l) ,

(2.20)

l

where the sum is over all α(l) = ( j1 , j2 , . . . , jλ ) such that |α(l)| = s and at most one ji ̸= j. The expression c(α(l), j, s) denotes a rational expression in terms of j, s and r j which is nonnegative whenever r j ≥ 1. Furthermore, define A j,α(l) = A j, j1 A j, j2 · · · A j, jλ . If A j,s ≤ 0 and r j ≥ 1 for all nonnegative integers j and all s ≥ j, Eq. (2.20) is equivalent to       A j+1,s = − |c(α(l), j, s)|  A j, j1   A j, j2  · · ·  A j, jλ  , (2.21) where the sum is over all α(l) = ( j1 , j2 , . . . , jλ ) such that |α(l)| = s and at most one ji ̸= j. Proof. Take the first term on the right side of Eq. (2.18), represent Akj, j A j,N as A j,α(l) , and 

r j +k−1 k r kj



as |c(α(l), j, s)|. Notice that (−1)k = (−1)λ(α(l))−1 . For the remaining terms on ˆ

the right side of Eq. (2.18) we combine them via Eq. (2.19), let Akj, j = A j,α(l) , and let 

|c(α(l), j, s)| =

ˆ r j +k−2 ˆ k−1 ˆ r k−1 j

 ˆ (r j −1)(k−1) . r j kˆ



Eq. (2.20) corresponds to Eq. (2.16) of [10].

27

H. Gingold, J. Quaintance / Journal of Approximation Theory 188 (2014) 19–38

An important consequence of Theorem 2.2 comes through repeated iterations of Eq. (2.21). We demonstrate what happens in the first summand on the right side of    A typical   iteration. Eq. (2.21) has the form |c(α(l), j, s)|  A j, j1   A j, j2  · · ·  A j, jλ . For each  A j, ji  we apply Eq. (2.21), with j → j − 1 and s → ji , to discover that  A j+1,s = − |c(α(l), j, s)| |c(α(l1 ), j1 , j1 )| |c(α(l2 ), j1 , j2 )| · · · |c(α(lλ ), j − 1, jλ )| |A j−1, jˆ1 | |A j−1, jˆ2 | · · · |A j−1, jˆp |, where jˆ1 + jˆ2 + · · · + jˆp = s. We may continue this process j − 1 times. In order to efficiently record the results let α = ( j1 , j2 , . . . , jn ) denote a vector with n components, each of which is a positive integer. Then aα(l) denotes the expression a j1 a j2 · · · a jn . After j iterations, assuming that A j,s ≤ 0 whenever s ≥ j, Eq. (2.21) becomes  A j+1,s = (−1)λ(α(l))+1 |c(α(l), j, s)|aα(l) l

=−



|c(α(l), j, s)| |a j1 | |a j2 | · · · |a jλ |,

(2.22)

l

where the sum is over all α(l) = ( j1 , j2 , . . . , jλ ) such that |α(l)| = s and c(α(l), j, s) is a j+1 rational expression in j, s, and {ri }i=1 which is nonnegative whenever ri ≥ 1. If s = j + 1 Eq. (2.22) becomes  A j+1, j+1 = r j+1 g j+1 = (−1)λ(α(l))+1 |c(α(l), j)|aα(l) l

=−



|c(α(l), j)| |a j1 | |a j2 | · · · |a jλ |,

(2.23)

l

where the sum is over all α(l) = ( j1 , j2 , . . . , jλ ) such that |α(l)| = j + 1. The following table explicitly demonstrates the structure of gi for 1 ≤ i ≤ 6. If ri ≥ 1 each coefficient is nonnegative. 1 r1 − 1 2 1 a1 , g2 = (−1)1 a1 + (−1)0 a2 r1 2r1r2 r2 2−1 r 1 1 g3 = (−1)2 1 2 a13 + (−1)1 a1 a2 + (−1)0 a3 r3 r3 3r1 r3 r2 − 1 2 1 + r1 (2r2 − 1) 2 g4 = (−1)1 a + (−1)2 a1 a2 + (−1)3 2r2r4 2 2r1r2r4 −2r2 + 2r13r2 − r13 + 2r12 − r1 4 1 1 × a1 + (−1)1 a1 a3 + (−1)0 a4 3 r4 r4 8r1 r2r4

g1 = (−1)0

1 2 1 1 1 1 a1 a3 + (−1)1 a2 a3 + (−1)2 a1 a22 + (−1)3 a13 a2 + (−1)1 a1 a4 r5 r5 r5 r5 r5 4−1 r 1 + (−1)4 1 4 a15 + (−1)0 a5 r5 5r1 r5

g5 = (−1)2

28

H. Gingold, J. Quaintance / Journal of Approximation Theory 188 (2014) 19–38

g6 = (−1)2

−r 2 + 3r 2 r3 + 1 1 2 1 r3 − 1 2 a1 a4 + (−1)1 a2 a4 + (−1)1 a3 + (−1)3 1 2 1 r6 r6 2r3 r6 3r1 r3 r6

2r3 − 1 a1 a2 a3 r3 r6 2 r −1 −r1 r3 + 3r1 r22 r3 − r1 r22 + r3 2 2 1 + (−1)2 2 2 a23 + (−1)3 a1 a2 + (−1)0 a6 r6 3r2 r6 2r1 r22 r3 r6

× a13 a3 + (−1)2

+ (−1)4 + (−1)5

4r22 − 4r12 r22 − 3r12 r3 + 6r1 r3 + 12r12 r22 r3 − 3r3 12r12 r22 r3 r6

a14 a2 + (−1)1

1 a1 a5 r6

12r15 r22 r3 − 9r13 r3 + 3r12 r3 − 12r22 r3 − 3r15 r3 + 9r3 r14 − 4r15 r22 + 8r22 r13 − 4r1 r22 72r15 r12 r3 r6

a16 .

We formalize the above discussion in the following lemma. It is a statement about a bijection between the sequence of the coefficients in a given power series and the sequence of coefficients in its GPPE expansion. Lemma 2.1. Let {rk }∞ k=1 denote a sequence ofnonzero complex numbers. Let gk ∈ C, k = 1, k rk 2, . . . , be an infinite sequence. Let the symbol ∞ k=1 (1 + gk x ) stand for the infinite product of elementary factors ∞ 

(1 + gk x k )rk := (1 + g1 x)r1 (1 + g2 x 2 )r2 · · · (1 + gk x k )rk · · · .

(2.24)

k=1

Then there exists a unique sequence an ∈ C, n = 1, 2, . . . , such that in the sense of power series the following holds 1+

∞  n=1

an x n :=

∞ 

(1 + gk x k )rk .

(2.25)

k=1

Conversely, let an ∈ C, n = 1, 2, . . . , be an infinite sequence. Then there exists a unique sequence of elements gk ∈ C, k = 1, 2, . . . , such that the identity (2.25) holds. Moreover, the elements gk have the representation provided by Eq. (2.23). Remark 2.1. It was shown in [12] that the non-positivity of the sequence of coefficients an ≤ 0, n = 1, 2, . . . is inherited by the sequence of coefficients gn ≤ 0, n = 1, 2, . . . . Our study shows that a myriad of other related sequences, namely A j,N ≤ 0, j = 2, 3, . . . , N = j + 1, j + 2, . . . inherit this non-positivity property. This peculiar and intriguing phenomenon extends also to the Taylor coefficients of the function log f (x). It is this representation of the coefficients A j,N as special polynomial functions of the coefficients an ≤ 0, n = 1, 2, . . . that underline the structure of the expansions given in this section. They will turn out to be central to convergence criteria that are taken up in the next section. It is noteworthy that the technique used by [12] in analyzing the signs of gk relies also on algorithms provided in [11]. The technique used in here may be described as more direct and explicit. 3. Convergence criterion for Eq. (2.1) Let {rn }∞ n=1 be a set of positive numbers such that rn ≥ 1 for all n. The structure of g j provided by Eq. (2.23) allows us to prove the following theorem.

H. Gingold, J. Quaintance / Journal of Approximation Theory 188 (2014) 19–38

Theorem 3.1. Let f (x) = 1 + GPPE f (x) = 1 +

∞ 

an x n =

n=1

∞

n=1 an x

∞ 

n

29

and {rn }∞ n=1 with rn ≥ 1 be given. Then f (x) has

(1 + gn x n )rn .

(3.1)

n=1

Consider the auxiliary functions C(x) = 1 −

∞ 

|an |x n =

n=1

M(x) = 1 −

∞ 

∞  

1 − Gˆ n x n

rn

,

(3.2)

n=1

Mn x n =

n=1

∞  

1 − En x n

rn

.

(3.3)

n=1

Assume that |an | ≤ Mn for all n. Then |gn | ≤ Gˆ n ≤ E n for all n. Proof. By Eq. (2.23) we have  gn = (−1)λ(α(l))+1 |c(α(l), n)|aα(l) l:|α(l)|=n

=

(−1)λ(α(l))+1 |c(α(l), n)|a j1 a j2 · · · a jλ ,



(3.4)

l:|α(l)|=n

since rn ≥ 1 is a nonnegative number. Eq. (3.4) implies that        λ(α(l)+1) |gn | =  (−1) |c(α(l), n)|a j1 a j2 · · · a jλ  l:|α(l)|=n   ≤ |c(α(l), n)| |a j1 | |a j2 | · · · |a jλ |.

(3.5)

l:|α(l)|=n

Eq. (2.23) when applied to Eq. (3.2) implies that  0 ≤ Gˆ n = (−1)λ(α(l)) |c(α(l), n)|(−|a j1 |)(−|a j2 |) · · · (−|a jλ |) l:|α(l)|=n

=



(−1)λ(2α(l)) |c(α(l), n)|(|a j1 |)(|a j2 |) · · · (|a jλ |)

l:|α(l)|=n

=



(3.6)

|c(α(l), n)| |a j1 | |a j2 | · · · |a jλ |.

l:|α(l)|=n

Combining Eqs. (3.5) and (3.6) shows that |gn | ≤ Gˆ n . Since |an | ≤ Mn we also have  0 ≤ Gˆ n = |c(α(l), n)| |a j1 | |a j2 | · · · |a jλ | l:|α(l)|=n

≤



|c(α(l), n)|M j1 M j2 · · · M jλ = E n ,

l:|α(l)|=n

where the last equality follows from Eq. (2.23). Thus Gˆ n ≤ E n .



Theorem 3.1, Part i, of [10] is now a special case of our Theorem 3.1.

30

H. Gingold, J. Quaintance / Journal of Approximation Theory 188 (2014) 19–38

Remark 3.1. Eqs. (2.22) allows for the following generalization of Theorem 3.1.  and (2.23) n and {r }∞ with r ≥ 1 be given. Consider the partial product Let f (x) = 1 + ∞ a x n n=1 n n=1 n expansion   p ∞ ∞    n n rn j f (x) = 1 + an x = (1 + gn x ) 1+ A p+1, j x . n=1

n=1

j= p+1

Also consider the partial product expansions of the auxiliary functions   p ∞ ∞    n n r j C(x) = 1 − |an |x = (1 − Gˆ n x ) n 1 − Aˆ p+1, j x ,

M(x) = 1 −

n=1

n=1

∞ 

p 

n

Mn x =

n=1

j= p+1

 (1 − E n x )

n rn

1−

n=1

∞ 

 E p+1, j x

j

.

j= p+1

  If |an | ≤ Mn for all n, we conclude that |gk | ≤ Gˆ k ≤ E k for 1 ≤ k ≤ p and that  A p+1, j  ≤ Aˆ p+1, j ≤ E p+1, j for all j ≥ p + 1. We now work with a particular case of M(x), namely M(x) = 1 −

∞ 

sn x n =

n=1

∞  

1 − En x n

rn

,

1

s ≡ sup |an | n .

(3.7)

n≥1

n=1

We want to determine when the GPPE of Eq. (3.7) will be absolutely convergent. Define ∞ (E n x n )l log(1− E n x n )rn := rn log (1 − E n x n ) := −rn l=1 , where we assume rn log (1 − E n x n ) l is well-defined if the series converges. Next define ∞ 

∞ ∞     (E n x n )l , rn log 1 − E n x n := − rn l n=1 n=1 l=1

(3.8)

by the convergence of the double series. Eq. (3.8) implies that if the double series is absolutely  ∞ (E n x n )l n n convergent, then both ∞ are n=1 rn log (1 − E n x ) and rn log (1 − E n x ) = −rn l=1 l absolutely convergent. Furthermore, the absolute convergence of the double series implies the  n rn absolute convergence of ∞ n=1 (1 − E n x ) since e

∞

n=1 rn

log(1−E n x n )

=e

∞

n=1 log(1−E n x

n )rn

=

∞  

1 − En x n

rn

.

n=1

 n Thus, it suffices to investigate the absolute convergence of ∞ n=1 rn log (1 − E n x ). ∞  ∞ rn n n Define log n=1 (1 − E n x ) := n=1 rn log (1 − E n x ) via Eq. (3.8). If we take the logarithm of Eq. (3.7) we find that   ∞ ∞     n n n rn log 1 − E n x = log 1 − s x . (3.9) n=1

n=1

Now 1−

∞  n=1

s n x n = 1 − sx

∞  n=0

(sx)n = 1 −

sx 1 − 2sx = . 1 − sx 1 − sx

H. Gingold, J. Quaintance / Journal of Approximation Theory 188 (2014) 19–38

31

Therefore,   ∞ ∞  (2sx)n  1 − 2sx (sx)n log = log(1 − 2sx) − log(1 − sx) = − + 1 − sx n n n=1 n=1 =

∞  1 − 2n n=1

n

(sx)n .

∞ 1−2n n By Ratio n=1 n (sx) absolutely converges whenever limn→∞  then+1  Test we know that  n(1−2 )  1 .  (n+1)(1−2n )  |sx| < 1. This is ensured by requiring |x| < 2s ∞ n We have shown that n=1 rn log (1 − E n x ) will be absolutely convergent whenever |x| < ∞ 1 n rn . n=1 (1 − E n x ) will also be absolutely converge for this range of x. We claim 2s Hence, this information provides a lower bound on the range of absolute convergence for the GPPE of Eq. (3.1). To determine the radiusof convergence for theGPPE of Eq. (3.1) we must determine ∞ n rn n the radius of convergence for log ∞ n=1 (1 + gn x ) := n=1 rn log(1 + gn x ), where the right side is defined via the convergence of the double series in Eq. (2.5). However,     ∞ ∞ ∞             (1 + gn x n )rn  =  rn log(1 + gn x n ) ≤ rn log(1 + gn x n ) log  n=1  n=1  n=1   ∞ ∞ ∞ ∞    (|gn | |x|n )k (−1)k−1 (gn x n )k    = rn  rn ≤  n=1 k=1 k=1 k k n=1 ≤

∞  ∞  n=1 k=1

rn

(E n |x|n )k , k

by Theorem 3.1.

 ∞ ∞ (E n x n )k n ,and hence ∞ These calculations imply that n=1 rn log (1 − E n x ), are n=1 k=1 rn k if ∞ ∞ n n r absolutely convergent, then n=1 rn log(1+gn x ) and n=1 (1+gn x ) n will also be absolutely convergence. We summarize our conclusions in the following theorem.  ∞ n Theorem 3.2. Let f (x) = 1 + ∞ n=1 an x and {rn }n=1 with rn ≥ 1 be given. 1

Define s := supn≥1 |an | n . Then both f (x) and its GPPE, f (x) = 1 +

∞ 

an x n =

n=1

∞ 

(1 + gn x n )rn ,

n=1

and the auxiliary function, along with its GPPE, M(x) = 1 −

∞  n=1

sn x n =

∞ 

(1 − E n x n )rn ,

(3.10)

n=1

will be absolutely convergent whenever |x| ≤

1 2s .

Theorem 3.2 corresponds to Theorem 2.5 of [12].  n Remark 3.2. The analysis that zero free regions of 1 + ∞ n=1 an x and the ra∞above shows n r dius of convergence of n=1 (1 + gn x ) n are intimately related. We bring out this point as yet another of the assumption  consequence ∞ ann n≤ 0, n = 1, 2, . . . . For a particular example take n x n = 1−2sx = 1 − −1 1+ ∞ a n=1 n=1 s x . This series has a radius of convergence ρ1 = s 1−sx

32

H. Gingold, J. Quaintance / Journal of Approximation Theory 188 (2014) 19–38

and a simple zero at x = (2s)−1 . We will show now that if we choose a sequence {rk }∞ k=1 with M rk ≥ 1, then n=1 (1 − E n x n )rn = 0 is impossible for any x with |x| < (2s)−1 and any inteM ger M. Observe that n=1 (1 − E n x n )rn = 0 if and only if 1 − E n x n = 0, i.e. if and only if E n x n = 1. We will show that E n x n < 1,

whenever 0 ≤ x ≤ (2s)−1 .

Consider Relation (2.10) written with gk = −E k , and dk = (1 − 2k+1 )s k+1 , namely, k+1  −(k + 1)rk+1 E k+1 = (1 − 2k+1 )s k+1 + nrn E n n . n|(k+1) n̸=k+1

This implies E k+1 (2s)−k−1 =

k+1  (2k+1 − 1)s k+1 (2s)−k−1 − [(k + 1)rk+1 ]−1 (2s)−k−1 nrn E n n (k + 1)rk+1 n|(k+1) n̸=k+1

(2k+1 − 1)s k+1 (2s)−k−1 (1 − 2−k−1 ) ≤ = < 1, (k + 1)rk+1 (k + 1)rk+1

k = 0, 1, 2, . . . ,

by virtue of the fact that −[(k + 1)rk+1 ]−1 (2s)−k−1



k+1

nrn E n n ≤ 0.

n|(k+1) n̸=k+1

∞ n n Evidently, 1−2sx ρ1 = s −1 and a simple zero at n=1 s x has a radius of convergence 1−sx = 1 −  n n x = (2s)−1 . It  follows from the above calculations that 1 − ∞ n=1 s x , and all the correspond∞ N ing series 1 + N = j A j,N x , j = 1, 2, . . . , have radius of convergence ρ j = ρ1 = s −1 and a simple zero at x = (2s)−1 . The remark above makes the following proposition self evident. Proposition 3.1. Let ρ j > 0 denote the radius of convergence of the series  N 1+ ∞ A j,N x , j = 1, 2, . . . . These series must diverge at x = ρ j if an ≤ 0, n = 1, 2, . . . . N= j M Assume that k=1 (1 + gk x k )rk ̸= 0 for |x| < ρ1 , M = 1, 2, . . . . Then ρ j = ρ1 , j = 1, 2, . . . . ∞ N = 0, j = n = 0 for some x, |x| < ρ . Then 1 + Moreover, let 1 + ∞ a x n 1 N = j A j,N x n=1 1, 2, . . . for the same value of x. Our next result provides an asymptotic formula for the majorizing GPPE of Eq. (3.10).  1−2sx n n Let {r }∞ be a sequence Theorem 3.3. Let f (x) = 1 − ∞ n=1 s x = 1−sx where s > 0. ∞ k k=1 k r such that rk ≥ 1. For this particular f (x) and its associated GPPE n=1 (1 + gk x ) k , we have rk (−gk ) v

(2k − 1)s k , k

(3.11)

k → ∞.

Proof. Start with Eq. (2.10) and rewrite it as (k + 1)rk+1 (−gk+1 ) = −dk − r1 (−g1 )k+1 −

 n|(k+1) k+1>n>1

nrn (−gn )

k+1 n

.

(3.12)

33

H. Gingold, J. Quaintance / Journal of Approximation Theory 188 (2014) 19–38

Since f (x) = 1 − f ′ (x)

∞

n=1 (sx)

n

=

1−2sx 1−sx

we discover that

(−2s)(1 − sx) + s(1 − 2sx) 1 − sx · f (x) 1 − 2sx (1 − sx)2 1 − sx −s −s · = = (1 − sx)(1 − 2sx) (1 − sx)2 1 − 2sx   ∞ ∞ ∞    −2s s k k k k k = + = s −2 2 s x + s x =s (−2k+1 + 1)s k x k . 1 − 2sx 1 − sx k=0 k=0 k=0 =

By definition

f ′ (x) f (x)

=

∞

k=0 dk x

k.

Hence, dk = (−2k+1 + 1)s k+1 , and Eq. (3.12) becomes 

(k + 1)rk+1 (−gk+1 ) = (2k+1 − 1)s k+1 − r1 (−g1 )k+1 −

nrn (−gn )

k+1 n

.

(3.13)

n|(k+1) k+1>n>1

 k k k Because f (x) = 1 − ∞ k=1 s x , and s > 0, the coefficient of x is negative whenever k ≥ 1. Eq. (2.23) in turn implies that −gk is positive for k ≥ 1. Denote the right side of Eq. (3.13) as T1 − T2 − ∆, where T1 := (2k+1 − 1)s k+1 ≥ 0, T2 := r1 (−g1 )k+1 ≥ 0, and  k+1 ∆ := n|(k+1) nr n (−gn ) n ≥ 0. Therefore k+1>n>1

(k + 1)rk+1 (−gk+1 ) = T1 − T2 − ∆ ≤ T1 = (2k+1 − 1)s k+1 = −dk .

(3.14)

We use Eq. (3.14) to provide an estimate for ∆. By definition  n  k+1   k+1 (2 − 1)s n n ∆= nrn nrn (−gn ) n ≤ nrn n|(k+1) n|(k+1) k+1>n>1

k+1 ≥n≥2 2



k+1 n

(nrn )

k+1 n

n



≤

(nrn )

(2s)

n|(k+1) k+1 ≥n≥2 2

1



≤ (2s)k+1

n|(k+1) k+1 ≥n≥2 2



n

k+1 n −1

= (2s)k+1

1

 n|(k+1) k+1 ≥n≥2 2

≤ (2s)k+1

(nrn ) 1

 k+1 2 ≥n≥2

n

 = (2s)k+1 

k+1 n −1

k+1 n −1

(3.15)

 1 2

k+1 2 −1

+

2 + k+1

1

 k+1 3 ≥n≥3

n

k+1 n −1

.

(3.16)

  k+1 n −1 = − k+1 − 1 ln n. Then and b(n, k) := − ln n k+1 n n n −1     ∂b(n, k) k+1 k+1 1 k+1 1 1 = ln n − − 1 = n − 1] + > 0, [ln ∂n n n n n k+1 n2

Define M :=



n ≥ 3.

k+1 3 ≥n≥3

1

(3.17)

Line (3.17) shows that b(n, k) is an increasing function with respect to n whenever n ≥ 3. Hence   k+1 k+1 k+1 b(n, k) < b , k = −(3 − 1) ln = −2 ln , 3 3 3

34

H. Gingold, J. Quaintance / Journal of Approximation Theory 188 (2014) 19–38

which implies each term of M satisfies eb(n,k) ≤ e−2 ln 1

 k+1 3 ≥n≥3

n

k+1 n −1

≤

 k+1 3 ≥n≥3

k+1 3

=

9 . (k+1)2

Therefore

9 9 9 ≤ (k + 1) = . 2 2 k+1 (k + 1) (k + 1)

(3.18)

Eq. (3.18) shows that limk→∞ M = 0. By combining  this result with Eq. (3.16) we conclude that limk→∞

∆

(2k+1 −1)s k+1

(2s)k+1 (1−2−k−1 )(2s)k+1 Since g1 = − rs1

1

= limk→∞

We now return to Eq. (3.14). rk+1 (−gk+1 ) =

2

k+1 −1 2

+

2 (k+1)

+ M = 0.

we discover that as k → ∞

T2 ∆ T1 − − k+1 k+1 k+1

 k+1 k+1 s r1 (2k+1 − 1)s k+1 r1 (−1) ∆ = − − k+1 k+1 k+1   k+1 k+1 k+1 (2 − 1)s (−1) ∆ = 1− k − k+1 r1 (2k+1 − 1) (2k+1 − 1)s k+1 =

(2k+1 − 1)s k+1 dk [1 + o(1)] = − [1 + o(1)] .  k+1 k+1

Also note the following. Remark 3.3. The asymptotic formula E k ∼ k −1 2k as k → ∞ for rk = 1, k = 1, 2, . . . obtained in [12] is now a special case of our result. Remark 3.4. Define a composition of an to be monomial of the form a j1 a j2 · · · a jm such that j1 + j2 + · · · + jm = n. By combining Eq. (2.4) with Theorem 3.3 we deduce that Eq. (3.11) −1 provides an upper bound on the number and weight of compositions of an whenever {|an |n }∞ n=1 −1 is monotone increasing sequence and s = limn→∞ |an |n . 4. A combinatorial interpretation for Eq. (2.1) and convergence Let n be a positive integer. A partition of n is a collection of positive integers whose sums equals n. For example the partitions of 4 are {4, 1 + 3, 2 + 2, 1 + 1 + 2, 1 + 1 + 1 + 1}. One of the most famous infinite products in mathematics is the generating function for partitions of distinct parts, namely ∞  n=1

(1 + x n ) = (1 + x)(1 + x 2 )(1 + x 3 ) · · · =

∞ 

pd (n)x n ,

(4.1)

n=0

where pd (n) counts the partition of n composed of distinct parts [2, Chapter 1]. By definition, pd (7) = 5 since {7, 1 + 6, 1 + 2 + 4, 3 + 4, 2 + 5} are the only partitions of 7 composed of distinct parts. Eq. (4.1) is the special case of Eq. (2.1) where gn = 1 and rn = 1. We can generalize Eq. (4.1) to encompass an arbitrary set of positive integers {rn }∞ n=1 if we define the notion of a partition of a multi-set. Identities involving partitions and compositions of multi-sets are found in the work of George Andrews, P. A. MacMahon, and J.J. Sylvester [1,3,4]. Historically, restrictions

H. Gingold, J. Quaintance / Journal of Approximation Theory 188 (2014) 19–38

35

are placed on the number of times a part may repeat in the multi-sets. We will not need to place such restrictions on our multi-sets. Let {rn }∞ n=0 be a set of nonnegative integers. Define the associated multi-set as 1r1 2r2 · · · k rk · · ·, where k rk denotes rk distinct copies of the integer k. If rk = 0 there are no copies of k in the multi-set. For example 12 24 33 45 denotes the multi-set {1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4}. For {rn }∞ n=0 a set of positive integers we form the generating function ∞ 

(1 + x n )rn = (1 + x)r1 (1 + x 2 )r2 · · · (1 + x k )rk · · · =

n=1

∞ 

pˆ d (n)x n ,

(4.2)

n=0

where pˆ d (n) counts the partitions of n with distinct parts composed from 1r1 2r2 · · · k rk · · ·. To clarify what is meant by distinct parts when working in the context of multi-sets, it helps to introduce the notion of color. Each of the rk copies of k is assigned a unique color from a set of rk colors. Differently colored k’s are considered distinct from each other. Thus pˆ d (n) counts the partitions of n over the multi-set 1r1 2r2 · · · k rk · · · which have distinct colored parts. As a case in point take the multi-set 12 24 33 45 . For this multi-set, r1 = 2, r2 = 4, r3 = 3, and r4 = 5. This means we have two colors for 1, Red (R) and Blue (B); four colors for 2, Red, Blue, Orange (O), and Yellow (Y); three colors for 3, Red, Blue, and Orange; and five colors for 4, Red, Blue, Orange, Yellow, and Green (G). The multi-set becomes {1 R , 1 B , 2 R , 2 B , 2 O , 2Y , 3 R , 3 B , 3 O , 4 R , 4 B , 4 O , 4Y , 4G } where the  color of the digit is denoted by the subscript. The generating function for this multi-set is 4n=1 (1 + x n )rn = (1 + x)2 (1 + x 2 )4 (1 + x 3 )3 (1 + x 4 )5 where exponent of x denotes the part while the exponent of each elementary factor denotes the number of colors available for the associated part. To calculate the partitions of 6 associated with this multi-set we calculate of 39 the coefficient n x 6 in the expansion of (1 + x)2 (1 + x 2 )4 (1 + x 3 )3 (1 + x 4 )5 = n=0 pˆ d (n)x . Such a partition, due to the two term structure of each polynomial factor, will consist of distinct colored parts. We find there are 62 partitions of 6 with distinct colored parts over the multiset {1 R , 1 B , 2 R , 2 B , 2 O , 2Y , 3 R , 3 B , 3 O , 4 R , 4 B , 4 O , 4Y , 4G }. In particular there are 5 partitions of the form 1 + 1 + 4; 6 partitions of the form 1 + 1 + 2 + 2; 24 partitions of the form 1 + 2 + 3; 4 partitions of the form 2 + 2 + 2; and 20 partitions of the form 2 + 4. In the case of 2 + 2 + 2 we have {2 R + 2 B + 2 O , 2 R + 2 B + 2Y , 2 R + 2 O + 2Y , 2 B + 2 O + 2Y }. Eq. (4.2) is Eq. (2.1) with gn = 1 and rn a positive integer. We can further generalize Eq. (4.2) by introducing collection of weights associated with each part, namely {gn }∞ n=1 , where gn is an arbitrary complex number. Eq. (4.2) becomes ∞  n=1

(1 + gn x n )rn = (1 + g1 x)r1 (1 + g2 x)r2 · · · (1 + gk x k )rk · · · =

∞ 

pˆ d (g, ¯ n)x n , (4.3)

n=0

α1 α2 αm where g¯is a finite polynomial in {gn }∞ n=1 , such that each monomial has the form g1 g2 · · · gm , m where i=1 iα1 = n and αm denotes the number of distinct colored copies of the part m which appear in the partition. For the multi-set  discussed previously, Eq. (4.3) becomes (1 + g1 x)2 (1 + g2 x 2 )4 (1 + g3 x 3 )3 (1 + g4 x 4 )5 = ¯ n)x n and the coefficient of x 6 n pˆ d (g, 3 2 2 is pˆ d (g1 , g2 , g3 , g4 , 6) = 5g1 g4 + 6g1 g2 + 24g1 g2 g3 + 4g2 + 20g2 g4 .

Eq. (4.3) provides a combinatorial interpretation for Eq. (2.1) whenever rn is a positive integer. We may also use colored multi-sets to provide a combinatorial interpretation whenever rn a

36

H. Gingold, J. Quaintance / Journal of Approximation Theory 188 (2014) 19–38

negative integer. First assume rn = −1 and gn = −1. Eq. (2.1) becomes ∞ 

(1 − x n )−1 = (1 − x)−1 (1 − x 2 )−2 (1 − x 3 )−1 · · · =

∞ 

p(n)x n ,

(4.4)

n=0

n=1

where p(n) is the number of partitions of n [2, Chapter 1]. To justify our interpretation of Eq. (4.4), we associate the exponent n in (1 − x n )−1 with the integer n. To determine the number of n’s used in a particular we expand by the geometric series (1 − x n ) = ∞ partition, n 2n 3n mn 1 + x + x + x + · · · = m=0 x , select a monomial, and divide the exponent by n. Assume rn is a positive integer. Eq. (4.4) may be generalized as ∞ 

(1 − x n )−rn = (1 − x)−r1 (1 − x 2 )−r2 (1 − x 3 )−r3 · · · =

n=1

∞ 

n p(n)x ˆ ,

(4.5)

n=0

where p(n) ˆ is the number of partitions of n associated with the colored multi-set which contains an unlimited repetition of each integer k in rk colors. The factor (1 − x k )−rk = (1 + x k + x 2k + x 3k + · · ·)rk corresponds to {k, k + k, k + k + k, . . .} replicated in rk colors. As an example, let r1 = 2, r2 = 1 and r3 = 3. The associated generating function is (1 − x)−2 (1 − x 2 )−1 (1 − x 3 )−3 . The multi-set we are working with contain two copies of {1, 1 + 1, 1 + 1 + 1, . . .}, one in Red and one in Blue; one copy of {2, 2 + 2, 2 + 2 + 2, . . .} in Red; and three copies of {3, 3 + 3, 3 + 3 + 3 · · ·} in Red, Blue, and Orange. To calculate the partitions of 5 we write (1 − x)−2 (1 − x 2 )−1 (1 − x 3 )−3 = (1 + x + x 2 + x 3 + · · ·)(1 + x + x 2 + x 3 + · · ·)(1 + x 2 + x 4 + x 6 + · · ·)(1 + x 3 + x 6 + x 9 + · · ·)(1 + x 3 + x 6 + x 9 + · · ·)(1 + x 3 + x 6 + x 9 + · · ·), expand, and collect all the monomials with x 5 . We get 18 since there are two ways to form 1 + 5, three ways to form 2 + 3, four ways to form 1 + 1 + 1 + 2, nine ways to form 1 + 1 + 3, and six ways to form 1 + 1 + 1 + 1 + 1, all of which we list below. 1+2+2 2+3 1+1+1+2 1+1+3 1+1+3 1+1+1+1+1 1+1+1+1+1

1R + 2R + 2R , 1B + 2R + 2R 2 R + 3 R , 2 R + 3B , 2 R + 3O 1R + 1R + 1R + 2R , 1R + 1R + 1B + 2R , 1R + 1B + 1B + 2R , 1B + 1B + 1B + 2R 1 R + 1B + 3 R , 1 R + 1B + 3B , 1 R + 1B + 3O , 1 R + 1 R + 3 R , 1R + 1R + 3B 1 R + 1 R + 3O , 1B + 1B + 3 R , 1B + 1B + 3B , 1B + 1B + 3O 1 R +1 R +1 R +1 R +1 R , 1 R +1 R +1 R +1 R +1 B , 1 R +1 R +1 R +1 B +1 B 1R + 1R + 1B + 1B + 1B , 1R + 1B + 1B + 1B + 1B , 1B + 1B + 1B + 1B + 1B

Eq. (4.5) can be further generalized by assigning a set of weights to each part. In particular we have ∞  (1 − gn x n )−rn = (1 − g1 x)−r1 (1 − g2 x 2 )−r2 (1 − g3 x 3 )−r3 · · · n=1

=

∞ 

p( ˆ g, ¯ n)x n ,

(4.6)

n=0

where g¯ is a polynomial in {gn }∞ colored parts i n=0 such that the exponent of gi tells the number of  αm m that appear in a partition of n. In other words, g¯ has the form g1α1 g2α2 · · · gm , where i=1 iα1 = n

H. Gingold, J. Quaintance / Journal of Approximation Theory 188 (2014) 19–38

37

and αm denotes the number of colored parts m which appear in the partition. For our previous example with r1 = 2, r2 = 1, and r3 = 3 we get (1 − g1 x)−2 (1 − g2 x 2 )−1 (1 − g3 x 3 )−3 = (1 + g1 x + g12 x 2 + g13 x 3 + · · ·)(1 + g1 x + g12 x 2 + g13 x 3 + · · ·)(1 + g2 x 2 + g2 x 4 + g23 x 6 + · · ·)(1 + g3 x 3 + g32 x 6 + g33 x 9 + · · ·)(1 + g3 x 3 + g32 x 6 + g33 x 9 + · · ·)(1 + g3 x 3 + g32 x 6 + g33 x 9 + · · ·). Then the coefficient of x 5 has the form 2g1 g22 + 3g2 g3 + 4g13 g2 + 9g12 g3 + 6g15 . Remark 4.1. It is noteworthy that the rate of growth of the counting coefficients pˆ d (n) and p(n) in (4.1) and (4.4) respectively, are important in combinatorial analysis. They are intimately related to the radius of convergence of the Taylor series in Eqs. (4.1) and (4.4). H. Gingold and A. Knopfmacher [12], proposed a convergence criterion that includes the numerous generating functions that occur in combinatorial analysis. The relevant theorem reads ∞ n Theorem 4.1 (Theorem 4.2 [12]). Let gl (z) = n=l gln z be an infinite sequence of formal power series with gln ≥ 0 for each l and n in N. Suppose that the formal product expansion f (z) = 1 +

∞  n=1

an z n =

∞ 

(1 + gl (z))

(4.7)

l=1

holds in the sense that  an = gl1 n 1 gl2 n 2 · · · glr nr ,

(4.8)

where the summation is over all (n 1 , n 2 , . . . , nr ) with n = n 1 + n 2 + · · · + nr , 1 ≤ n 1 ≤ n 2 ≤ · · · ≤ nr and 1 ≤ l1 < l2 < · · · < lr . ∞  n (i) If l=1 gl (z) is absolutely convergent for 0 ≤ z < R, then 1 + ∞ n=1 an z with coefficients satisfying (4.8) converges in |z| < R. (ii) If f (z) is ananalytic function in |z| < R then for each l, gl (z) converges absolutely in ∞ |z| < R and l=1 (1 + gl (z)) converges absolutely in |z| < R to f (z). Here are two pertinent examples which demonstrate the usefulness of Theorem ∞ ∞ l4.1. First take rk = gk = 1 for all k. Then gl (x) = x l . Evidently, l=1 gl (x) = l=1 x is absolutely convergent inside the unit disk. Theorem 4.1 implies that the radius of convergence ∞ ∞ sl of k is 1. For the second example let r = g = −1. Then g (x) = 1 + p (k)x k k l s=1 x and ∞ k=1 d ∞ ∞ sl g (x) = x is absolutely convergent inside the unit disk. Once again, Theol=1 l l=1 s=1  k converges inside the unit disk. In fact, Theorem 2 [14], rem 4.1 implies that 1 + ∞ p(k)x k=1 which involves generating functions of partitions, is a special case of Theorem 4.1. References [1] George E. Andrews, Rogers–Ramanujan identities for two-color partitions, Indian J. Math. 29 (2) (1987) 117–125. Ramanujan Centenary Volume. [2] George E. Andrews, Theory of Partitions, in: Cambridge Mathematical Library, Cambridge University Press, Cambridge, UK, 1998. [3] George E. Andrews, Theory of compositions IV; mutlicompositions, Math. Stud. Special Centenary Volume (2007) 25–31. [4] George E. Andrews, A survey of mutipartitions: Congruences and identities, in: Surveys in Number Theory, Springer Science, 2008. [5] S. Borofsky, Expansions of functions defined by Dirichlet series into infinite series and products, Tohoku Math. J. 34 (1930) 263–274. [6] S. Borofsky, Expansion of analytic functions into infinite products, Ann. of Math. 32 (1931) 23–36. [7] J.M. Feld, The expansion of analytic functions in generalized Lambert series, Ann. of Math. 33 (1932) 139–142.

38

H. Gingold, J. Quaintance / Journal of Approximation Theory 188 (2014) 19–38

[8] J.M. Feld, P. Newman, On the representation of analytic functions of several variables as infinite products, Bull. Amer. Math. Soc. 36 (1930) 284–288. [9] H. Gingold, A note on the reduction of operations via power product approximations, Util. Math. 37 (1990) 79–90. [10] H. Gingold, Factorization of matrix functions and their inverses via power product expansion, Linear Algebra Appl. 430 (2008) 2835–3140. [11] H. Gingold, H.W. Gould, Michael E. Mays, Power product expansions, Util. Math. 34 (1988) 143–161. [12] H. Gingold, A. Knopfmacher, Analytic properties of power product expansions, Canad. J. Math. 47 (6) (1995) 1219–1239. [13] H. Gingold, A. Knopfmacher, D.S. Lubinsky, The zero distribution of the partial products of power expansions, Analysis 13 (1993) 133–157. [14] E. Grosswald, Topics from the Theory of Numbers, second ed., Birkh¨auser, 1984. [15] G.H. Hardy, A note on the continuity or discontinuity of a function defined by an infinite product, Proc. Lond. Math. Soc. (2) 7 (1909) 40–48. [16] F. Hertzog, Remarks on a paper by Kolberg, Nord. Math. Tidskr. 10 (1962) 78–79. [17] H. Indlekofer, R. Warlimont, Remarks on the infinite product representations of holomorphic functions, Publ. Math. Debrecen 41 (1992) 263–276. [18] P.W. Ketchum, Review of [6], Jahrb. Fortshritte Math. 58 (1932) 323–324. [19] A. Knopfmacher, Generalized series expansions of functions, in: P. Nevai, A. Pinkus (Eds.), Progress in Approximation Theory, Academic Press, 1991, pp. 513–533. [20] A. Knopfmacher, Infinite product factorization of analytic functions, J. Math. Anal. Appl. 162 (1991) 526–536. [21] A. Knopfmacher, L. Lucht, The radius of convergence of power product expansions, Analysis 11 (1991) 91–99. [22] A. Knopfmacher, A.M. Odlyzko, B. Pittel, B. Richmond, D. Stark, G. Szekeres, N. Wormald, The asymptotic number of set partitions with unequal block sizes, Electron. J. Combin. 6 (1999) 1–36. [23] A. Knopfmacher, J.N. Ridely, Reciprocal sums over partitions and compositions, SIAM J. Discrete Math. 6 (1993) 388–399. [24] O. Kolberg, A property of the coefficients in a certain product expansion of the exponential function, Nord. Math. Tidskr. 10 (1960) 33–34. [25] J.F. Ritt, Representation of analytic functions in infinite product expansions, Math. Z. 32 (1930) 1–3.