Morphological and histochemical characterization of the mucosa of the digestive tract in Engraulis anchoita.

Introduction and statement of results

Nicolas [ 14 ] and DeSalvo–Pak [ 7 ] proved that p ( n ), the number of partitions of n , is log-concave for n ≥ 26 and even n ≥ 2 . That is,

This follows for large enough n directly from the Hardy–Ramanujan asymptotic expansion for p ( n ) (see [ 7 , §6.2]), but log-concavity for n ≥ 26 requires a careful argument. DeSalvo and Pak proceeded from Rademacher’s exact formula for p ( n ), together with work of Lehmer [ 10 , 11 ]. There has since been a flurry of results studying log-concavity and higher Turán inequalities for partition generating functions and weakly holomorphic modular forms (see for example [ 6 , 8 , 9 , 15 ]).

In this paper, we consider unimodal sequences, which distinguish themselves from the other examples by their connection to false theta functions. Let u ( n ) count the number of unimodal sequences of n ,

DeSalvo and Pak [ 7 ] mentioned that asymptotic formulas for unimodal sequences (or stacks/convex compositions) are not precise enough to prove log-concavity. We overcome this problem, using recent work of the second author and Nazaroglu, proving an exact formula for u ( n ). Here, we work out precise asymptotic expansions with explicit error term.

Throughout, we write f ( n ) = O ≤ c ( g ( n ) ) if | f ( n ) | ≤ c | g ( n ) | and set n 0 : = 100 000 .

Theorem 1.1

For n ≥ n 0 , we have

where the constants A , B , C , D , and E are defined in equation ( 4.16 ).

Remark

Presumably both the constant in the error term as well as the number of terms may be improved significantly with our methods. We only stop at the power n - 2 to obtain log-concavity.

We expect our method applies to other exact formulas for coefficients of mixed mock/false modular objects. Indeed, Mauth in [ 13 ] follows this approach to prove log-concavity for so-called partitions without sequences considered by the authors in [ 2 ]. Another example arises from irreducible characters of certain vertex operator algebras considered by Cesana [ 5 ].

The following asymptotic expansion is a direct consequence of Theorem 1.1 .

Corollary 1.2

For n ≥ n 0 , we have

Combined with a numerical check (see the remark in Section 2), log-concavity follows.

Corollary 1.3

For n ∈ N \ { 1 , 5 , 7 } , we have

For n ∈ { 1 , 5 , 7 } , the above difference is negative.

Remark

Note that a combinatorial proof of the log-concavity for integer partitions is still open. Indeed, as there are so many failures of log-concavity (i.e., for odd 1 ≤ n ≤ 25 ), this is likely a difficult problem. The number of exceptions for unimodal sequences by contrast is much more manageable, so perhaps it is reasonable to ask for a combinatorial proof in this case.

The paper is organized as follows. In Sect. 2 , we recall the exact formula for u ( n ) from [ 4 ] and state some inequalities for the I -Bessel function. The proof of Theorem 1.1 is carried out in Sects. 3 and 4 : In Sect. 3 , we bound the contribution to u ( n ) of the terms for k ≥ 2 in Theorem 2.1 , in Sect. 4 , we use the saddle-point method to prove an asymptotic expansion for the term k = 1 in Theorem 2.1 , finishing the proof of Theorem 1.1 .

Preliminaries

Recall that the generating function for unimodal sequences is given by (see [ 1 , equation (3.2) and (3.3)])

Remark

Recall that ( q ; q ) ∞ - 1 is the partition generating function. The right-hand side of ( 2.1 ) allows for a quick computation of u ( n ) as a convolution of pairs of partitions and the coefficients { 0 , ± 1 } in the sparse series. This is especially true with programs like Wolfram Mathematica which have the partitions of large order already hard-coded.

In [ 4 ], we found the following exact formula for 1 u ( n ).

Theorem 2.1

([ 4 ], Theorem 1.3, negative of the second term) We have

where I 3 2 is the I -Bessel function of order 3 2 and K k ( n , r ) is a certain Kloostermann-type sum (in particular | K k ( n , r ) | ≤ k ).

We require the following bounds for the I 3 2 -Bessel function; the proof uses the integral representation of [ 12 , page 172] and [ 16 , equation (6.25)]. See also [ 3 , Lemma 2.2].

Lemma 2.2

We have

The terms k ≥ 2

In this section we bound the contribution from k ≥ 2 in Theorem 2.1 . We begin by estimating the sum on r .

Lemma 3.1

For | x | ≤ 1 and k ≥ 2 , we have

Proof

We use for 0 ≤ y ≤ π

to bound

A short calculation shows that for | x | ≤ 1 ,

Thus, using | K k ( n , r ) | ≤ k , we have

The claim now follows from the integral comparison criterion. □

Now we bound the sum over from all k ≥ 2 in Theorem 2.1 as follows.

Lemma 3.2

For n ≥ n 0 , we have

Proof

By Lemma 3.1 , we can bound the left-hand side by

To estimate the integral of the I 3 2 -Bessel function, we apply Lemma 2.2 .

First we consider the range

Applying Lemma 2.2 and extending the range of integration, we have

Using the evaluation

this part contributes overall

Next we consider the range

If k ≥ π 24 n + 1 3 2 , then this range is empty and we have no contribution. Otherwise, Lemma 2.2 gives that the corresponding contribution to ( 3.1 ) can be bound against

We now trivially bound this against the exponential

where in the last step we use n ≥ n 0 . Combining ( 3.2 ) and ( 3.3 ) proves the lemma. □

The main term and the proof of Theorem 1.1

The term from k = 1 in Theorem 2.1 equals

Using

we obtain that ( 4.1 ) equals

We bound the second term in ( 4.2 ) for n ≥ n 0 as

We split the integral for the first term in ( 4.2 ) as

We bound the contribution from the second range as

The rest of this section is devoted to obtaining an asymptotic expansion for

of the form in Theorem 1.1 . That is, our error term needs to take the shape

In keeping with the saddle-point method, we eventually let x ↦ c n - 1 4 x for some constant c , so d x ↦ c n - 1 4 d x , and thus we obtain the factor O ( n - 5 4 ) outside of the integral. Hence, we need to expand the integrand itself up to O ( n - 5 2 ) . To that end, we begin with the following lemma. Set

Lemma 4.1

For | x | ≤ n - 1 8 , we have

Proof

By expanding the Taylor series for 1 - x 2 and noting that | x | ≤ n - 1 8 ≤ 1 2 , we have

Now using that e u ≤ 1 + 2 | u | for u ∈ [ - 1 , 1 ] and | x | ≤ n - 1 8 , we have

Next, we note that

so that in particular | y | ≤ 1.3 for | x | ≤ n - 1 8 . Hence,

The lemma follows. □

Next, we require the Taylor expansions of the other functions in the integrand in ( 4.4 ), namely

where P odd ( x ) is an odd polynomial, so does not contribute to the integral, and also

Thus, by Lemma 4.1 and equations ( 4.5 ) and ( 4.6 ), we see that ( 4.4 ) equals

Set λ n : = ( π 6 2 24 n + 1 ) 1 2 . It is not hard to see that

Next we make the change of variables x ↦ x λ n in ( 4.7 ) to get

Now note that

where for | X | ≤ 0.5

Thus 2 F ( x λ n ) = O ≤ 0.06 ( x 10 λ n 10 ) .

We now plug ( 4.10 ) into ( 4.9 ) and simplify (using a computer algebra system as aide) to get

where E n ( x ) contains only powers λ n - m with m ≥ 10 . We bound the integral of E n explicitly using a computer algebra system to carry out the integral and arrive at

Now we use that for m even with 0 ≤ m ≤ 16 ,

We now use for w ∈ R ≥ 1 and m even with 0 ≤ m ≤ 16 ,

Define

With ( 4.8 ), we have

We use ( 4.13 ) in ( 4.11 ), simplify and combine with ( 4.12 ). Thus ( 4.11 ) equals

Finally, we apply to ( 4.14 ) the Taylor expansions for n ≥ n 0

Thus ( 4.14 ) equals

Define

We now combine ( 4.15 ) with Lemma 3.2 and ( 4.3 ). From

we conclude Theorem 1.1 .

Acknowledgements

We are grateful to Lukas Mauth for helping us verify log-concavity for the remaining cases 7 ≤ n ≤ n 0 with Wolfram Mathematica. We thank the referee for providing feedback which improved the exposition. The authors have received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 101001179).

Funding

Open Access funding enabled and organized by Projekt DEAL.

Data availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

Footnotes

Note that if u ∗ ( n ) denotes the unimodal sequences counted in [ 4 ], then u ( n ) = coeff [ q n ] 1 ( q ; q ) ∞ 2 - u ∗ ( n ) (see [ 2 , footnote on page 4]).

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Associated Data

Data Availability Statement

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.