An expansion for the number of partitions of an integer
Stella Brassesco, Arnaud Meyroneinc
TL;DR
This paper derives an asymptotic expansion for the partition function p(n), using probabilistic methods and cumulant expansions to improve understanding of its growth as n becomes large.
Contribution
It introduces a novel asymptotic expansion for p(n) based on characteristic functions and cumulants, providing explicit formulas and a concise leading factor.
Findings
Derived an asymptotic series in inverse powers of √n
Provided explicit formulas for cumulants and the leading factor
Achieved a more accurate approximation for p(n) as n grows large
Abstract
We obtain an asymptotic expansion for , the number of partitions of a natural number , starting from a formula that relates its generating function with the characteristic functions of a family of sums of independent random variables indexed by . The expansion consists of a factor (which is the leading term) times an asymptotic series expansion in inverse powers of a quantity that grows as as , and whose coefficients are simple combinatorial expressions. The asymptotic series is obtained by expanding the characteristic functions in terms of the cumulants of the random variables, for which simple and accurate approximations are derived, as well as explicit exact formulae. That computations also give a concise expression for the factor.
| () | |||
|---|---|---|---|
| 10 | 42 | 1 | |
| 11 | 57 | 1.0178571428571428571 | |
| 50 | 0.9999265519571455152 | ||
| 51 | 1.0000666825037613100 | ||
| 100 | 0.9999981791400054107 | ||
| 101 | 1.0000017390807618196 | ||
| 200 | 0.9999999908879917331 | ||
| 201 | 1.0000000086963580162 | ||
| 500 | 0.9999999999997561054 | ||
| 600 | 0.9999999999999841826 | ||
| 700 | 0.9999999999999987223 | ||
| 800 | 0.9999999999999998772 | ||
| 900 | 0.9999999999999999863 | ||
| 1000 | 0.9999999999999999983 |
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An expansion for the number of partitions of an integer
Stella Brassesco
Arnaud Meyroneinc
Abstract
We consider the number of partitions of a natural number , starting from an expression derived by L. Báez-Duarte in [1] by relating its generating function with the characteristic functions of a family of sums of independent random variables indexed by . The asymptotic formula for follows then from a local central limit theorem as suitably with . We take further that analysis and compute formulae for the terms that compose that expression, which accurately approximate them as . Those include the generating function and the cumulants of the random variables. After developing an asymptotic series expansion for the integral term we obtain an expansion for that can be simplified as follows: for each ,
[TABLE]
The coefficients are positive and have simple expressions as finite sums of combinatorial numbers, and the remainder satisfies as .
The cumulants are given by series of rational functions and the approximate formulae obtained could be also of independent interest in other contexts.
Departamento de Matemáticas, Instituto Venezolano de Investigaciones Científicas, Apartado Postal 20632 Caracas 1020–A, Venezuela
[email protected] [email protected]
Dedicated to the memory of Luis Báez Duarte
Keywords: Integer partitions, Asymptotic expansions, Asymptotic formulae, Central limit theorem, Cumulants.
11P82 (05A16 05A17)
1 Introduction
For each , consider the partition number of , that is, the number of non increasing positive integer sequences such that the sum of its terms adds to . The study of partitions goes back to L. Euler, who established in particular the generating function of the sequence ,
[TABLE]
In a celebrated article published a century ago [8], G.H. Hardy and S. Ramanujan considered the order of magnitude of as , and obtained an asymptotic series expansion which in particular yields the asymptotic formula
[TABLE]
H. Rademacher in [9] considered a modification of the asymptotic series that turns it convergent, and yields a value for that differs by decimals from its exact value when taking of the order of terms for large .
Much more recently, J.H. Bruinier and K. Ono in [4] deduced an exact formula for that expresses it as a finite sum of algebraic quantities. In [6] M. Dewar and R. Murty developed this formula to derive an asymptotic expansion in the spirit of that of Rademacher.
L. Báez Duarte in [1] considered a formula for , that, when approximating some of the terms with the aid of the Euler–Maclaurin summation formula, yields (1.2) as a consequence of a local central limit theorem.
With that formula as starting point, we develop an asymptotic expansion for , consisting of a factor growing exponentially in , times an asymptotic series expansion given in terms of inverse powers of a certain quantity that is very nearly a rational function of growing as , while the coefficients are explicitly given as sums of combinatorial numbers. Although the series is shown to be convergent for any , it is really an asymptotic expansion, with errors of the order as when taking terms.
These results are derived from an essentially real variable analysis, avoiding the difficulties due to the complicated structure of the function near the border of the unit circle in . The final expansion can be simplified to obtain an explicit expression in terms of rational functions of .
We compare the approximations resulting by truncating the simplified expansion with the true value of for several values of , to illustrate the results.
2 Preliminaries and statement of results
As discussed in [1] (see also the references therein), the series in (1.1) determines for each a random variable defined on some probability space and taking values on by taking {\mathbb{P}}\big{(}X(t)=n\big{)}=\frac{p(n)\,t^{n}}{f(t)}. We denote by the probability and by the expectation with respect to . The characteristic function of is directly computed:
[TABLE]
Each factor in (1.1) determines analogously a random variable taking values on the multiples of :
[TABLE]
whose characteristic function is . It follows then that the random variable can be expressed as the infinite sum of the independent random variables . Moreover, is a random variable with geometric distribution with parameter . In other words,
[TABLE]
with independent geometric random variables. The mean and variance of are directly computed,
[TABLE]
On the other hand (2.1) may be rewritten in terms of the sequence
[TABLE]
which yields by Fourier inversion
[TABLE]
Observe that the series (2.3) and (2.4) are convergent for , and that both and as . We obtain indeed the precise asymptotic behaviour as of and in Corollary 3.5. Let us define then the sequence
[TABLE]
and note that as . Substitute and change variables in the integral above to obtain
[TABLE]
where is obtained by suitably centering and scaling :
[TABLE]
and we have denoted as usual by the standard deviation of . From (2.2), (2.3), (2.4) and (2.5), the central limit theorem implies that is asymptotically normal (as ). Moreover, in [1], it is proved that a local central limit theorem holds and the integral above converges to , what yields after determining the asymptotics of the terms in (2.6) as the asymptotic formula (1.2).
We obtain accurate approximations (as ) for the terms in the factor, and an asymptotic series expansion for the integral term in (2.6), following by expanding the integrand in terms of the cumulants of the random variables. The resulting quantities do not seem to have simple expressions in terms of , but we show that they are well approximated by elementary functions. Indeed, let us define
[TABLE]
The expansion that we obtain turns out to be given in terms of , which is very well approximated (as ) by . Precisely, it follows from Lemma 4.1 that for some positive constant ,
[TABLE]
Most of our formulae are given in terms of , that is shown in Corollary 3.6 to satisfy:
[TABLE]
In terms of those quantities, we can now state our main results.
Theorem 2.1**.**
For any given , satisfies
[TABLE]
where:
The coefficients are given by
[TABLE]
and satisfy .
The error satisfies as .
It is possible to clear up the expression (2.11) by replacing and by the approximations that follow from (2.9) and (2.10). After verifying that the errors introduced by doing that are sufficiently small to be included in the error terms of the expansion, the following alternative expression is obtained.
Proposition 2.2**.**
For any given , satisfies
[TABLE]
where:
The coefficients are those in (2.12).
The error satisfies as .
The expansion (2.13) is indeed less precise than that in Theorem 2.1, but its terms are very simple to implement. We report in Table 1 a comparison of the approximate values for obtained by computing the sum in (2.13) with for several values of . The results appear reasonably accurate even for the small values of that were considered.
Both statements are proved in Section 5, after developing asymptotic formulae for each of the terms in (2.6).
In Section 3 we consider the terms in the factor. The behaviour of as is derived from a functional formula, which by means of a recurrence permits also to deduce simple formulae for the first two cumulants that approximate them very accurately as . Those in turn render very good approximations for the sequence . That analysis turns out the exponential term in (2.11).
In Section 4 we obtain an asymptotic series expansion for the integral term in (2.6), which produces the sum in the expansion of . The proof is based on an expansion of the integrand in terms of the cumulants of the random variables. In the Appendix, we obtain functional formulae for all those cumulants, and deduce from them their precise asymptotics as , which in turn permits to control the errors in the expansion. We derive as well some properties of the cumulant generating function needed in the analysis. Explicit formulae for all the cumulants of , are also obtained, similar to those for the first two cumulants in (2.3) and (2.4) and whose numerators are given in terms of Eulerian polynomials.
3 Asymptotic formulae for the terms in the factor
In this section, we obtain expressions for the terms in the factor in (2.6), suitable for determining their behaviour for large , which corresponds to close to . The asymptotics of as is derived from the functional formula for stated in the next result.
Lemma 3.1**.**
For ,
[TABLE]
Proof.
The following identity for , is Corollary (ii) in [2, p.256]:
[TABLE]
The substitution and correspondingly in the above product yields from (1.1) an expression for , that after taking logarithm and reordering is exactly (3.1). ∎
Let us observe now that, as , the last term in (3.1) goes to zero. Moreover, starting from (1.1) we can estimate for ,
[TABLE]
to obtain, as an application of the mean value theorem that for some
[TABLE]
In particular, if we take any and then , and we conclude that
[TABLE]
For instance, taking we obtain that, for any ,
[TABLE]
We may thus regard (3.1) as an asymptotic formula for as , which we write in the form
[TABLE]
with satisfying (3.2). We also have from the above estimations that . The symbol denotes asymptotic logarithmic order, as formalized in the following definition. We find this kind of asymptotic equivalence along the way in the sequel.
Definition 3.2**.**
We say that a function defined for in the interval is logarithmically equivalent to for some as and denote
[TABLE]
if
[TABLE]
We proceed to deduce formulae for the mean and variance suitable for the analysis as , and used in Corollary 3.6 to derive precise estimates for as from (2.5). We recall that and are given by the series (2.3) and (2.4), as already computed in the previous section,
[TABLE]
Lemma 3.3**.**
The mean and variance satisfy the following functional equations,
[TABLE]
Remark 3.4**.**
The first series in (3.5) can be computed explicitly in the case , as follows from formula (8.3), page 255 of B.C. Berndt’s edition of S. Ramanujan’s notebooks [2]:
[TABLE]
Indeed, the identity (3.6) follows from that formula as well. An account of series like those in (3.5), that appear also in other contexts, can be found in the references therein. Some examples are mentioned in [7]. A closed form of for other values of does not appear to be known.
The formulae in the previous lemma yield precise asymptotics for and as , as we state next.
Corollary 3.5**.**
[TABLE]
where
[TABLE]
Before proving Lemma 3.3 and the above corollary, let us recall the notion of cumulants of a random variable and some of their properties.
Given a random variable with characteristic function , its cumulants (or semi-invariants) are the coefficients in the Taylor series expansion of (provided such expansion exists),
[TABLE]
We consider here and in the sequel the principal value of the logarithm. Therefore, and is referred to as the cumulant generating function of the random variable . The first cumulant results to be the mean, and the second one the variance, when those exist. We refer to [10] for further properties of .
In the case of the random variables , considered in the introduction, from (2.1)
[TABLE]
and it is not difficult to verify from this last expression that its cumulants, that we denote by , are finite and can be computed from the simple recursion formulae
[TABLE]
Starting from
[TABLE]
we recover from the first identity the series in (3.5) for and . Another couple of forms for those series follows from the second identity; we recall them here for further reference:
[TABLE]
It is not difficult to conclude from (3.5) and the recurrence (3.11) that, besides being finite for each , the cumulants , satisfy as . Explicit formulae for are computed in the Appendix.
Proof of Lemma 3.3.
Observe that for any real function with derivative , if ,
[TABLE]
whenever both sides are defined. Apply next to (3.1) to obtain formula (3.6) directly from (3.11), and again to (3.6) to obtain (3.7). ∎
Proof of Corollary 3.5.
Note that from Lemma 3.3, (3.8) and (3.9),
[TABLE]
It is thus enough to show that, given , there are positive constants and such that if ,
[TABLE]
The lower bounds follow by estimating the series (3.5) by their first term, as all the terms are positive, and the denominator is bounded away from zero if . The upper bound follows from the mean value theorem as in the proof of (3.2). Indeed, for , observe that, from (3.12),
[TABLE]
and then, for any and ,
[TABLE]
In particular, if we choose , we can see that for any ,
[TABLE]
The same procedure yields the upper bound in (3.16) for . ∎
We deduce next good approximations (as ) to convenient multiples of and its reciprocal, in terms of the rational function already defined in (2.8). Recall that was defined in (2.5) as the solution to , and it is therefore an increasing sequence converging to 1 as .
Corollary 3.6**.**
The parameter introduced in (2.5) satisfies
[TABLE]
Moreover, there is a positive constant such that
[TABLE]
Proof.
[TABLE]
Picking only the positive terms on the left hand side we obtain
[TABLE]
which yields the lower bound in (3.18), while the upper bound is a direct consequence of the definition (2.5). Since increases with , from (3.18)
[TABLE]
thus the inequality (3.17) holds for any when taking . Furthermore, from (3.22)
[TABLE]
and then from (3.17), for each
[TABLE]
As decreases with , we also have that
[TABLE]
Denote and multiply (3.21) by to obtain that satisfies
[TABLE]
Observe that so it indeed depends on , but we know from (3.25) that it is positive and exponentially small. Solving the equation we obtain
[TABLE]
In terms of , let us write
[TABLE]
with
[TABLE]
To estimate the difference in (3.19), from (3.27) we write
[TABLE]
whence, from (3.29)
[TABLE]
and the proof of (3.19) follows from (3.25) and the definition of . The proof of (3.20) follows as well from (3.27) and the above estimates. ∎
As consequence of the previous analysis, we obtain an expression for the factor in (2.6), that yields precise asymptotics as .
Corollary 3.7**.**
The factor in front of the integral term in (2.6) satisfies
[TABLE]
where satisfies that, for some positive constant
[TABLE]
Proof.
Let us write the factor in (2.6) as
[TABLE]
From (3.4) evaluated at , (3.21) and the definition of (3.25) it follows after collecting terms that
[TABLE]
The second identity follows from (3.27). Next, observe that if we write the square root as in (3.28), after expanding we obtain
[TABLE]
Let us express, with the aid of Corollary 3.6
[TABLE]
Substitution of the last two formulae into (3.32) yields
[TABLE]
with .
But we know from (3.3) and (3.24) that
[TABLE]
which, together with (3.25) yields (3.31). ∎
Remark 3.8**.**
Good approximations to in terms of follow from Corollary 3.5 and (3.19). In particular, the blunt estimate
[TABLE]
is obtained. We prefer to keep that term untouched here, as some simplifications occur when considering the integral term in the next section, and more precise estimates are presented there.
4 The expansion of the integral term
Let us turn now to consider the integral in (2.6). In [1], it is proved that
[TABLE]
which, together with Corollary 3.7 and (3.33) yields the asymptotic formula (1.2) from (2.6).
We follow the approach of [3] to obtain an asymptotic series expansion for the above integral. The analysis here is more involved, among other reasons due to the fact that the terms are defined implicitly, and we need to consider approximations. We develop the integrand in terms of the cumulants of the random variables . In the appendix, we obtain explicit expressions for each of the cumulants as series of rational functions of , whose numerators are given in terms of Eulerian polynomials. Simple functional formulae similar to those already obtained in the previous section for the first two cumulants and are also deduced from the recurrence relation (3.11). They yield accurate approximations (as ), which permit to control the errors. The expansion results to be given in inverse powers of , with defined in Section 2:
[TABLE]
It is easy to see from Corollary 3.5 and (3.19) that . It is indeed very nearly , which was defined in (2.8), as precisely established in the following lemma.
Lemma 4.1**.**
The sequence satisfies that, for some positive constant ,
[TABLE]
Proof.
From (3.9), (3.27), (3.15) and (3.25),
[TABLE]
with as defined in (3.29). From (3.19), (3.16) and (3.24) we conclude that (4.3) holds. ∎
Theorem 4.2**.**
Given , the integral in (4.1) can be expanded as
[TABLE]
where:
The coefficients are
[TABLE]
and they satisfy
[TABLE]
The error term satisfies
[TABLE]
Proof.
To obtain the asymptotics, it is convenient to split the integration interval into disjoint regions and , according to or not for some positive constant :
[TABLE]
The value of is fixed hereafter and it is not included in the notation.
Let us consider first the integral in , starting with
[TABLE]
We follow the computations in the proof of the theorem in [1, p.119] to estimate the integrand. From (2.1) and (1.1),
[TABLE]
where the sum is estimated by taking only its first term and then recalling that the functions and are increasing for . Whence, if ,
[TABLE]
Finally, take in the above estimates and recall that, as observed in (3.23), for all to conclude
[TABLE]
which, together with (3.9), (3.18) and (3.19) implies that, for some positive constants and we have
[TABLE]
To consider the integral over we expand the integrand in (4.1) in terms of the cumulants. To that end, observe that, for , we may write
[TABLE]
with the cumulant generating function of defined in the previous section. It is clear from the definition (2.7) that the first cumulant of is zero, while the cumulants of order for are , where are the cumulants of the random variable given by (3.11). For each , we consider the Taylor expansion of order for . From Lemma A.3,
[TABLE]
where the remainder can be estimated with the aid of Corollary A.4:
[TABLE]
We substitute the expressions for the cumulants obtained in Proposition A.2 into (4.11) to write
[TABLE]
where
[TABLE]
If we deduce from (A.11) that there is a positive constant that may depend on such that, if ,
[TABLE]
and that the series resulting from taking in the sum in (4.13) converges to
[TABLE]
Thus, from (4.13) we may write, for :
[TABLE]
where
[TABLE]
It follows that for some constant that depends only on ,
[TABLE]
Observe next that if , then , so (4.16) is valid when evaluated at and , as well as the estimates (4.12), (4.15) and (4.18). Recall the definition of (4.2), and let us denote
[TABLE]
[TABLE]
to write, from (3.9) and (4.16):
[TABLE]
for
[TABLE]
for some positive constant , as follows from (3.15), (3.16), (3.19) and (3.24).
We proceed now to obtain a suitable expansion for the last line in (4.21) following the procedure in [3]. For each ,
[TABLE]
In the case , the remainder satisfies
[TABLE]
Substitute the above expansion with in (4.21) and integrate over to obtain that for each
[TABLE]
where is the error term arising from (4.21):
[TABLE]
It is easy to see that, for and ,
[TABLE]
where is a polynomial of degree with coefficients depending on products of derivatives (up to order ) of the functions and . Those derivatives are directly seen to be bounded if . We can write then the first integral in the right hand side of (4.24) as the difference of the corresponding one on the whole line and that over . We denote the latter by to obtain:
[TABLE]
Observe that all the integrals above are finite. Moreover, we can compute the integral in . Indeed, recall that and . Substitute into the integrand in (4.27), interchange the derivative with the integral and integrate term by term using that
[TABLE]
Take, for the given , to obtain from (4.24) and (4.27) after renaming the terms in the sum that
[TABLE]
where
[TABLE]
Observe now that from (4.19) and Lemma 4.1
[TABLE]
with . The derivatives in (4.28) are easily computed: for ,
[TABLE]
Thus, substitution into (4.28), after expressing in terms of and clearing up yields the expansion on the right hand side of (4.5). From (4.9), to finish the proof of Theorem 4.2 it suffices to show that:
- a)
It is possible to choose such that satisfies (4.8).
- b)
.
*Proof of *a) We estimate each of the terms in the last line of (4.28), starting with , that was defined in (4.25). Observe first that if ,
[TABLE]
The inequality follows from observing that and that, for , so from (4.20)
[TABLE]
To estimate the errors that compose , we use the following inequality, that is a direct consequence of Proposition A.2 and (A.11): for each , there is a positive constant that may depend on such that
[TABLE]
Next, we choose and decompose the region of integration as the disjoint union for
[TABLE]
For we obtain from (4.12) and (4.18), with the aid of (4.32):
[TABLE]
From these inequalities plus (4.15) evaluated at we conclude that, given , if , we can take sufficiently large such that there is a constant
[TABLE]
In the case , again from (4.12) and (4.18) and with the aid of (4.32) we write
[TABLE]
Recall that and that from (3.7), as , thus the ’s, can be taken smaller that for sufficiently large by choosing large. In addition, for , from the definition (4.22) of , from (4.20) and using the estimate (A.11) for we conclude that, for each given there is a positive constant such that
[TABLE]
Then, for large enough, from (4.33), (4.34), (4.35), (4.15), the last inequality above and (4.31) we estimate as follows:
[TABLE]
Since the above integrals are finite and with , we conclude with the aid of (3.20) that, given , there exists such that, for sufficiently large there are positive constants , such that
[TABLE]
It remains to consider , defined in (4.29).
For , from (4.23), (4.26) and the remark below, using (4.31) we obtain
[TABLE]
It follows then that, for some polynomial with bounded coefficients (depending on the derivatives of and over ),
[TABLE]
and, for some constant ,
[TABLE]
Since , it follows from the above estimates that given ,
[TABLE]
what concludes the proof of a).
*Proof of *b) We proceed to show that the ’s defined in (4.30) alternate sign. Define, for each fixed , , and . It is easy to verify that for any and and then,
[TABLE]
In particular, the last formula implies that , and from the identity in (4.6), we have
[TABLE]
thus . Let us show now that . Since it follows that if is odd,
[TABLE]
and if is even
[TABLE]
which is summarized as
[TABLE]
and from (4.38)
[TABLE]
what concludes the proof of b), and that of Theorem 4.2. ∎
Remark 4.3**.**
The inequalities (4.7) in particular imply that the sum in (4.5) is convergent (as ) for any such that , which is the case for any , as follows for instance from (4.3). However, (4.5) is an asymptotic expansion, and has to be interpreted as such: for fixed , it approximates the integral on the left hand side for large, with an error satisfying (4.8).
5 Proof of the main results and a table
We present here the proofs of Theorem 2.1 and Proposition 2.2. The expansion obtained in this last result is computed for several values of , taking , and compared with the true value of , as reported in Table 1.
Proof of Theorem 2.1.
From (2.6) and Corollary 3.7, we know that
[TABLE]
Let us denote by the last line above. For sufficiently large we can estimate with the aid of (3.31), after recalling that and that as ,
[TABLE]
Substituting the integral by the expansion obtained in Theorem 4.2 we have that the second line in (5.1) equals
[TABLE]
Thus, (2.11) follows after substituting the last expression into (5.1), collecting terms after expressing from (4.2) and denoting by the terms coming from and , that are directly seen to have the required order. Observe that the coefficients are those computed in Theorem 4.2, and the stated inequalities are precisely (4.7). ∎
Proof of Proposition 2.2.
For each , let us estimate the difference of the sums in (2.13) and (2.11) as follows:
[TABLE]
where we have denoted and used that for all ; we have also assumed that is large enough so that (recall ) so the series is convergent, as can be seen from (4.7). The last inequality is a consequence of (4.3), with a positive constant.
To estimate the difference of the factors in (2.13) and (2.11), observe first that, after simple computations using (4.2) and (3.9) we have
[TABLE]
Thus, from (3.20), and estimating with the aid of (3.15), (3.16) and (3.24), we obtain that, for a positive constant,
[TABLE]
Let us then add and substract \frac{2\pi^{2}\,{\rm e}^{r_{n}}}{3\sqrt{3}(1+2r_{n})^{2}}\big{(}1-\sum_{\ell=1}^{N}\frac{D_{\ell}}{(1+2\,r_{n})^{\ell}}\big{)} to the expression (2.11) for to obtain
[TABLE]
To conclude, call the expression following the first sum above, and observe that, from the above estimates for and , it is easy to conclude that it has the same order as . ∎
Appendix A Formulae for higher order cumulants
In the next lemma, we derive a couple of explicit expressions for each cumulant , in terms of the Eulerian polynomials , that are defined through the following identity
[TABLE]
The first four are:
[TABLE]
More details can be found in L. Comtet’s book [5].
Lemma A.1**.**
The cumulants satisfy
[TABLE]
The above series are absolutely convergent for .
Proof.
Denote by the right hand side of (A.3). In (3.12) we have seen that , which equals by (A.2); that proves (A.3) for . To conclude, from (3.11) it suffices to show that
[TABLE]
Using a well known recurrence relation for the Eulerian polynomials that can be found for instance in [5, p.292]
[TABLE]
we compute
[TABLE]
Summing up in the last expression we obtain , and conclude the proof of (A.3). To prove (A.4), observe that, from (A.3) and (A.1), if ,
[TABLE]
The convergence of the series in (A.3) and (A.4) is clear from the fact that the polynomials for any . ∎
The next result is a consequence of Lemma A.1 and a reasoning similar to that leading to prove Lemma 3.3: from the recurrence (3.11) we derive functional equations for for , that yield asymptotic formulae as for those. The precise statement is given next.
Proposition A.2**.**
The cumulants , satisfy the following functional equations
[TABLE]
where the terms are given by the following expression
[TABLE]
and satisfy
[TABLE]
Proof.
Recall that the are obtained from the recurrence (3.11). We already know from Lemma 3.3 and Corollary 3.5 that the proposition holds for . Let us start from as given in (3.7), and denote the -th iteration of the operator . Use (3.13) with the first two terms in the right hand side of (3.7) to obtain directly by induction that for ,
[TABLE]
which are the first two terms on the right hand side of (A.7). Thus, the terms also satisfy the recurrence
[TABLE]
To prove (A.8) for it is enough to verify that the expression satisfies the above recurrence, which follows by a straightforward induction in .
[TABLE]
Since is a polynomial in for each , the above series is easily seen to be bounded from above by a constant if , say, so we conclude from the mean value theorem that, if (and then ),
[TABLE]
In addition recall that the Eulerian polynomials have non negative coefficients, and coefficient one in the linear terms for , to obtain from (A.3) by just taking the first term in each series the following lower bounds:
[TABLE]
Thus, (A.9) holds. Indeed, from (A.10) and (A.8) we conclude that there is a positive constant that may depend on such that for any ,
[TABLE]
∎
We computed in Lemma A.1 the derivatives at zero of the cumulant generating function . To estimate the remainder in the Taylor formula we need also the derivatives of at . Recall that
[TABLE]
Lemma A.3**.**
For each , ,
[TABLE]
where the functions are defined for by formula (A.3):
[TABLE]
Proof.
Observe first that, from the same reasoning used to prove (A.3), the functions satisfy the recurrence . From the expression in terms of , it is straightforward to see by differentiating the series that
[TABLE]
Differentiating the last expression we obtain, using the recurrence for the ’s
[TABLE]
which is the desired expression for . Successive differentiation using the recurrence yields the general formula. ∎
Corollary A.4**.**
The derivatives of satisfy
[TABLE]
for some positive constant that may depend on .
Proof.
From Lemma A.3,
[TABLE]
where to estimate the denominators we use that , and to estimate de numerators it is enough to observe that the Eulerian polynomials have real positive coefficients. The right hand side above is precisely that in item a) in the statement. To prove the inequality b), we write, from (A.7) for and :
[TABLE]
For each , we take above and use (A.11) and (3.19) to conclude that b) in the statement holds. ∎
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