# Asymptotics of Plancherel measures for symmetric groups

## Abstract

We consider the asymptotics of the Plancherel measures on partitions of as goes to infinity. We prove that the local structure of a Plancherel typical partition in the middle of the limit shape converges to a determinantal point process with the discrete sine kernel.

On the edges of the limit shape, we prove that the joint distribution of suitably scaled 1st, 2nd, and so on rows of a Plancherel typical diagram converges to the corresponding distribution for eigenvalues of random Hermitian matrices (given by the Airy kernel). This proves a conjecture due to Baik, Deift, and Johansson by methods different from the Riemann-Hilbert techniques used in their original papers and from the combinatorial proof given by the second author.

Our approach is based on an exact determinantal formula for the correlation functions of the poissonized Plancherel measures in terms of a new kernel involving Bessel functions. Our asymptotic analysis relies on the classical asymptotic formulas for the Bessel functions and depoissonization techniques.

## 1. Introduction

### 1.1. Plancherel measures

Given a finite group by the corresponding Plancherel measure we mean the probability measure on the set , of irreducible representations of which assigns to a representation the weight For the symmetric group . the set , is the set of partitions of the number which we shall identify with Young diagrams with , squares throughout this paper. The Plancherel measure on partitions arises naturally in representation–theoretic, combinatorial, and probabilistic problems. For example, the Plancherel distribution of the first part of a partition coincides with the distribution of the longest increasing subsequence of a uniformly distributed random permutation Reference31.

We denote the Plancherel measure on partitions of by ,

where is the dimension of the corresponding representation of The asymptotic properties of these measures as . have been studied very intensively; see the References and below.

In the seventies, Logan and Shepp Reference23 and, independently, Vershik and Kerov Reference40Reference42 discovered the following measure concentration phenomenon for as Let . be a partition of and let and be the usual coordinates on the diagrams, namely, the row number and the column number. Introduce new coordinates and by

that is, we flip the diagram, rotate it as in Figure 1, and scale it by the factor of in both directions.

After this scaling, the Plancherel measures converge as (see Reference23Reference40Reference42 for precise statements) to the delta measure supported on the following shape:

where the function is defined by

The function is plotted in Figure 1. As explained in detail in Reference22, this limit shape is closely connected to Wigner’s semicircle law for distribution of eigenvalues of random matrices; see also Reference19Reference20Reference21.

From a different point of view, the connection with random matrices was observed in Reference3Reference4, and also in earlier papers Reference16Reference28Reference29. In Reference3, Baik, Deift, and Johansson made the following conjecture. They conjectured that in the limit and after proper scaling the joint distribution of , becomes identical to the joint distribution of properly scaled largest eigenvalues of a Gaussian random Hermitian matrix (which form the so-called Airy ensemble; see Section ,1.4). They proved this for the individual distribution of and in Reference3 and Reference4, respectively. A combinatorial proof of the full conjecture was given by one of us in Reference25. It was based on an interplay between maps on surfaces and ramified coverings of the sphere.

In this paper we study the local structure of a typical Plancherel diagram both in the bulk of the limit shape and on its edge, where by the study of the edge we mean the study of the behavior of , and so on. ,

We employ an analytic approach based on an exact formula in terms of Bessel functions for the correlation functions of the so-called *poissonization* of the Plancherel measures (see Theorem 1 in the following subsection), and the so-called *depoissonization* techniques (see Section 1.4).

The exact formula in Theorem 1 is a limit case of a formula from Reference8; see also the recent papers Reference26Reference27 for more general results. The use of poissonization and depoissonization is very much in the spirit of Reference3Reference16Reference39 and represents a well–known statistical mechanics principle of the equivalence of canonical and grand canonical ensembles.

Our main results are the following two. In the bulk of the limit shape we prove that the local structure of a Plancherel typical partition converges to a determinantal point process with the discrete sine kernel; see Theorem ,3. This result is parallel to the corresponding result for random matrices. On the edge of the limit shape, we give an analytic proof of the Baik-Deift-Johansson conjecture; see Theorem 4. These results will be stated in Sections 1.3 and 1.4 of the present Introduction, respectively.

Simultaneously and independently, results equivalent to our Theorems 2 and 4 were obtained by K. Johansson Reference17.

### 1.2. Poissonization and correlation functions

For consider the ,*poissonization* of the measures :

This is a probability measure on the set of all partitions. Our first result is the computation of the correlation functions of the measures .

By correlation functions we mean the following. By definition, set

Also, following Reference41, define the *modified Frobenius coordinates* of a partition by

where stands for the symmetric difference of two sets, is the number of squares on the diagonal of and , and ’s are the usual Frobenius coordinates of ’s Recall that . is the number of squares in the row to the right of the diagonal, and th is number of squares in the column below the diagonal. The equality thEquation1.2 is a well–known combinatorial fact discovered by Frobenius; see Ex. I.1.15(a) in Reference24. Note that, in contrast to the set , is infinite and, moreover, it contains all but finitely many negative integers.

The sets and have the following nice geometric interpretation. Let the diagram be flipped and rotated as in Figure 1, but not scaled. Denote by a piecewise linear function with whose graph is given by the upper boundary of completed by the lines

Then

In other words, if we consider as a history of a walk on then , are those moments when a step is made in the negative direction. It is therefore natural to call the *descent set* of As we shall see, the correspondence . is a very convenient way to encode the local structure of the boundary of .

The halves in the definition of have the following interpretation: one splits the diagonal squares in half and gives half to the rows and half to the columns.

#### Definition 1.1

The correlation functions of are the probabilities that the sets or, similarly, contain a fixed subset More precisely, we set .

#### Theorem 1

For any we have

where the kernel is given by the following formula:

The functions are defined by

where is the Bessel function of order and argument The diagonal values . are determined by the l’Hospital rule.

This theorem is established in Section 2.1; see also Remark 1.2 below. By the complementation principle (see Sections A.3 and 2.2), Theorem 1 is equivalent to the following

#### Theorem 2

For any we have

Here the kernel is given by the following formula:

where The diagonal values . are determined by the l’Hospital rule.

#### Remark 1.2

Theorem 1 is a limit case of Theorem 3.3 of Reference8. For the reader’s convenience a direct proof of it is given in Section 2. Various limit cases of the results of Reference8 are discussed in Reference9. By different methods, the formula Equation1.8 was obtained by K. Johansson Reference17.

A representation–theoretic proof of a more general formula than Theorem 3.3 of Reference8 has been subsequently given in Reference27Reference26; see also Reference7.

#### Remark 1.3

Observe that all Bessel functions involved in the above formulas are of integer order. Also note that the ratios like are entire functions of and because is an entire function of In particular, the values . are well defined. Various denominator–free formulas for the kernel are given in Section 2.1.

### 1.3. Asymptotics in the bulk of the spectrum

Given a sequence of subsets

where is some fixed integer, we call this sequence *regular* if the limits

exist, finite or infinite. Here Observe that if . is finite, then for .

In the case when can be represented as and the distance between and goes to as we shall say that the sequence *splits*; otherwise, we call it *nonsplit*. Obviously, is nonsplit if and only if all stay at a finite distance from each other.

Define the correlation functions of the measures by the same rule as in Equation1.4:

We are interested in the limit of as This limit will be computed in Theorem .3 below. As we shall see, if splits, then the limit correlations factor accordingly.

Introduce the following *discrete sine kernel* which is a translation invariant kernel on the lattice ,

depending on a real parameter :

Note that and for we have

where is the Tchebyshev polynomials of the second kind. We agree that

and also that

The following result describes the local structure of a Plancherel typical partition.

#### Theorem 3

Let be a regular sequence and let the numbers , be defined by Equation1.10, Equation1.11. If splits, that is, if and the distance between and goes to as then ,

If is nonsplit, then

where is the discrete sine kernel and .

We prove this theorem in Section 3.

#### Remark 1.4

Notice that, in particular, Theorem 3 implies that, as the shape of a typical partition , near any point of the limit curve is described by a stationary random process. For distinct points on the curve these random processes are independent.

#### Remark 1.5

By complementation (see Sections A.3 and 3.2), one obtains from Theorem 3 an equivalent statement about the asymptotics of the following correlation functions:

#### Remark 1.6

The discrete sine kernel was studied before (see Reference44Reference45), mainly as a model case for the continuous sine kernel. In particular, the asymptotics of Toeplitz determinants built from the discrete sine kernel was obtained by H. Widom Reference45 who was answering a question of F. Dyson. We thank S. Kerov for pointing out this reference.

#### Remark 1.7

Note that, in particular, Theorem 3 implies that the limit density (the 1-point correlation function) is given by

This is in agreement with the Logan-Shepp-Vershik-Kerov result about the limit shape More concretely, the function . is related to the density Equation1.14 by

which can be interpreted as follows. Approximately, we have

Set Then the above relation reads . and it should be satisfied on the boundary of the limit shape. Since we conclude that ,

as was to be shown.

#### Remark 1.8

The discrete sine kernel becomes especially nice near the diagonal, that is, where Indeed, .

### 1.4. Behavior near the edge of the spectrum and the Airy ensemble

The discrete sine kernel vanishes if Therefore, it follows from Theorem .3 that the limit correlations vanish if for some However, as will be shown below in Proposition .4.1, after a suitable scaling near the edge the correlation functions , converge to the correlation functions given by the Airy kernel Reference12Reference36

Here is the Airy function:

In fact, the following more precise statement is true about the behavior of the Plancherel measure near the edge By symmetry, everything we say about the edge . applies to the opposite edge .

Consider the random point process on whose correlation functions are given by the determinants

and let

be its random configuration. We call the random variables the ’s*Airy ensemble*. It is known Reference12Reference36 that the Airy ensemble describes the behavior of the (properly scaled) 1st, 2nd, and so on largest eigenvalues of a Gaussian random Hermitian matrix. The distribution of individual eigenvalues was obtained by Tracy and Widom in Reference36 in terms of certain Painlevé transcendents.

It has been conjectured by Baik, Deift, and Johansson that the random variables

converge, in distribution and together with all moments, to the Airy ensemble. They verified this conjecture for individual distribution of and in Reference3 and Reference4, respectively. In particular, in the case of this generalizes the result of ,Reference40Reference42 that in probability as The computation of . was known as the Ulam problem; different solutions to this problem were given in Reference1Reference16Reference32; see also the survey Reference2.

Convergence of all expectations of the form

to the corresponding quantities for the Airy ensemble was established in Reference25. The proof in Reference25 was based on a combinatorial interpretation of Equation1.16 as the asymptotics in a certain enumeration problem for random surfaces.

In the present paper we use different ideas to prove the following

#### Theorem 4

As the random variables , converge, in joint distribution, to the Airy ensemble.

This is done in Section 4 using methods described in the next subsection. The result stated in Theorem 4 was independently obtained by K. Johansson in Reference17. See, for example, Reference13 for an application of Theorem 4.

### 1.5. Poissonization and depoissonization

We obtain Theorems 3 and 4 from Theorem 1 using the so-called depoissonization techniques. We recall that the fundamental idea of depoissonization is the following.

Given a sequence its *poissonization* is, by definition, the function

Provided the grow not too rapidly this is an entire function of ’s In combinatorics, it is usually called the exponential generating function of the sequence . Various methods of extracting asymptotics of sequences from their generating functions are classically known and widely used (see for example .Reference39 where such methods are used to obtain the limit shape of a typical partition under various measures on the set of partitions).

A probabilistic way to look at the generating function Equation1.17 is the following. If then , is the expectation of where is a Poisson random variable with parameter Because . has mean and standard deviation one expects that ,

provided the variations of for are small. One possible regularity condition on which implies Equation1.18 is monotonicity. In a very general and very convenient form, a depoissonization lemma for nonincreasing nonnegative was established by K. Johansson in Reference16. We use this lemma in Section 4 to prove Theorem 4.

Another approach to depoissonization is to use a contour integral

where is any contour around Suppose, for a moment, that . is constant, The function . has a unique critical point If we choose . as the contour then only neighborhoods of size , contribute to the asymptotics of Equation1.19. Therefore, for general we still expect that provided the overall growth of , is under control and the variations of for are small, the asymptotically significant contribution to Equation1.19 will come from That is, we still expect .Equation1.18 to be valid. See, for example, Reference15 for a comprehensive discussion and survey of this approach.

We use this approach to prove Theorem 3 in Section 3. The growth conditions on which are suitable in our situation are spelled out in Lemma 3.1.

In our case, the functions are combinations of the Bessel functions. Their asymptotic behavior as can be obtained directly from the classical results on asymptotics of Bessel functions which are discussed, for example, in the fundamental Watson’s treatise Reference43. These asymptotic formulas for Bessel functions are derived using the integral representations of Bessel functions and the steepest descent method. The different behavior of the asymptotics in the bulk of the spectrum, near the edges of the spectrum, and outside of is produced by the different location of the saddle point in these three cases.

### 1.6. Organization of the paper

Section 2 contains the proof of Theorems 1 and 2 and also various formulas for the kernels and We also discuss a difference operator which commutes with . and its possible applications.

Section 3 deals with the behavior of the Plancherel measure in the bulk of the spectrum; there we prove Theorem 3. Theorem 4 and a similar result (Theorem 5) for the poissonized measure are established in Section 4.

At the end of the paper there is an Appendix, where we collected some necessary results about Fredholm determinants, point processes, and convergence of trace class operators.

## 2. Correlation functions of the measures

### 2.1. Proof of Theorem 1

As noted above, Theorem 1 is a limit case of Theorem 3.3 of Reference8. That theorem concerns a family of probability measures on partitions of where , are certain parameters. When the parameters go to infinity, tends to the Plancherel measure Theorem 3.3 in .Reference8 gives a determinantal formula for the correlation functions of the measure

in terms of a certain *hypergeometric kernel*. Here and is an additional parameter. As and the negative binomial distribution in ,Equation2.1 tends to the Poisson distribution with parameter In the same limit, the hypergeometric kernel becomes the kernel . of Theorem 1. The Bessel functions appear as a suitable degeneration of hypergeometric functions.

Recently, these results of Reference8 were considerably generalized in Reference26, where it was shown how this type of correlation functions can be computed using simple commutation relations in the infinite wedge space.

For the reader’s convenience, we present here a direct and elementary proof of Theorem 1 which uses the same ideas as in Reference8 plus an additional technical trick, namely, differentiation with respect to which kills denominators. This trick yields a denominator–free integral formula for the kernel see Proposition ;2.7. Our proof here is a verification, not a derivation. For more conceptual approaches the reader is referred to Reference26Reference27Reference7.

Let Introduce the following kernel . :

We shall consider the kernels and as operators in the space on .

We recall that simple multiplicative formulas (for example, the hook formula) are known for the number in Equation1.1. For our purposes, it is convenient to rewrite the hook formula in the following determinantal form. Let be the Frobenius coordinates of see Section ;1.2. We have

The following proposition is a straightforward computation using Equation2.2.

#### Proposition 2.1

Let be a partition. Then

where are the modified Frobenius coordinates of defined in Equation1.2.

Let be the push-forward of under the map Note that the image of . consists of sets having equally many positive and negative elements. For other the right-hand side of ,Equation2.3 can be easily seen to vanish. Therefore is a determinantal point process (see the Appendix) corresponding to that is, its configuration probabilities are determinants of the form ,Equation2.3.

#### Corollary 2.2

.

This follows from the fact that is a probability measure. This is explained in Propositions A.1 and A.4 in the Appendix. Note that, in general, one needs to check that is a trace class operator.Footnote^{1} Actually, is of trace class because the sum of the absolute values of its matrix elements is finite. We are grateful to P. Deift for this remark.^{✖} However, because of the special form of it suffices to check a weaker claim – that , is a Hilbert–Schmidt operator, which is immediate.

Theorem 1 now follows from general properties of determinantal point processes (see Proposition A.6 in the Appendix) and the following

#### Proposition 2.3

.

We shall need three identities for Bessel functions which are degenerations of the identities (3.13–15) in Reference8 for the hypergeometric function. The first identity is due to Lommel (see Reference43, Section 3.2, or Reference14, 7.2.(60)):

The other two identities are the following.

#### Lemma 2.4

For any and any we have

#### Proof.

Another identity due to Lommel (see Reference43, Section 5.23, or Reference14, 7.15.(10)) reads

Substituting we get Equation2.5. Substituting yields

Let be the difference of the left-hand side and the right-hand side in Equation2.6. Using Equation2.7 and the recurrence relation

we find that Hence for any . it is a periodic function of and it suffices to show that Clearly, the left-hand side in .Equation2.6 goes to 0 as From the defining series for . it is clear that

which implies that the right-hand side of Equation2.6 also goes to as This concludes the proof. .

#### Proof of Proposition 2.3.

It is convenient to set Since the operator . is invertible we have to check that

This is clearly true for therefore, it suffices to check that ;

where and Using the formulas .

one computes

where Similarly, .

Now the verification of Equation2.10 becomes a straightforward application of the formulas Equation2.5 and Equation2.6, except for the occurrence of the singularity in those formulas. This singularity is resolved using Equation2.4. This concludes the proof of Proposition 2.3 and Theorem 1.

### 2.2. Proof of Theorem 2

Recall that by construction

Let us check that this and Proposition A.8 imply Theorem 2. In Proposition A.8 we substitute

By definition, set

We have the following

#### Lemma 2.5

.

It is clear that since the cancel out of all determinantal formulas, this lemma and Proposition -factorsA.8 establish the equivalence of Theorems 1 and 2.

#### Proof.

Using the relation

and the definition of one computes

Clearly, the relation Equation2.12 remains valid for It remains to consider the case . In this case we have to show that .

Rewrite it as

By Equation2.14 this is equivalent to

Examine the right-hand side. The terms with vanish because then The term with . is equal to 1, which corresponds to 1 in the left-hand side. Next, the terms with vanish because for these values of the expression , vanishes. Finally, for set , Then the . term in the second sum is equal to minus the th term in the first sum. Indeed, this follows from the trivial relation th

This concludes the proof.

### 2.3. Various formulas for the kernel

Recall that since is an entire function of the function , is entire in and We shall now obtain several denominator–free formulas for the kernel . .

#### Proposition 2.6

#### Proof.

Straightforward computation using a formula due to Nielsen (see Section 5.41 of Reference43 or Reference14, formula 7.2.(48)).

#### Proposition 2.7

Suppose Then .

#### Proof.

Follows from a computation done in the proof of Proposition 2.3,

and the following corollary of Equation2.14:

#### Remark 2.8

Observe that by Proposition 2.7 the operator is a sum of two operators of rank 1.

#### Proposition 2.9

#### Proof.

Our argument is similar to an argument due to Tracy and Widom; see the proof of the formula (4.6) in Reference36. The recurrence relation Equation2.8 implies that

Consequently, the difference between the left-hand side and the right-hand side of Equation2.15 is a function which depends only on Let . and go to infinity in such a way that remains fixed. Because of the asymptotics Equation2.9 both sides in Equation2.15 tend to zero and, hence, the difference actually is 0.

In the same way as in Reference36 this results in the following

#### Corollary 2.10

For any the restriction of the kernel , to the subset defines a nonnegative trace class operator in the space on that subset.

#### Proof.

By Proposition 2.9, the restriction of on is the square of the kernel Since the latter kernel is real and symmetric, the kernel . is nonnegative. Hence, it remains to prove that its trace is finite. Again, by Proposition 2.9, this trace is equal to

This sum is clearly finite by Equation2.9.

#### Remark 2.11

The kernel resembles a Christoffel–Darboux kernel and, in fact, the operator in defined by the kernel is an Hermitian projection operator. Recall that where , is of the form

One can prove that this together with Lemma 2.5 imply that is an Hermitian projection kernel. However, in contrast to a Christoffel–Darboux kernel, it projects to an infinite–dimensional subspace.

Note that in Reference17 the restriction of the kernel to was obtained as a limit of Christoffel–Darboux kernels for Charlier polynomials.

### 2.4. Commuting difference operator

Consider the difference operators and on the lattice ,

Note that as operators on Consider the following second order difference Sturm–Liouville operator: .

where and are operators of multiplication by certain functions , The operator .Equation2.17 is self–adjoint in A straightforward computation shows that .

It follows that if for a certain then the space of functions , vanishing for is invariant under .

#### Proposition 2.12

Let denote the operator in obtained by restricting the kernel to Then the difference Sturm–Liouville operator .Equation2.17 commutes with provided

#### Proof.

Since is the square of the operator with the kernel it suffices to check that the latter operator commutes with , with the above choice of , and But this is readily checked using .Equation2.18.

This proposition is a counterpart of a known fact about the Airy kernel; see Reference36. Moreover, in the scaling limit when and

the difference operator becomes, for a suitable choice of the constant, the differential operator

which commutes with the Airy operator restricted to The above differential operator is exactly that of Tracy and Widom .Reference36.

#### Remark 2.13

Presumably, this commuting difference operator can be used to obtain, as was done in Reference36 for the Airy kernel, asymptotic formulas for the eigenvalues of where , and Such asymptotic formulas may be very useful if one wishes to refine Theorem .4 and to establish convergence of moments in addition to convergence of distribution functions. For individual distributions of and the convergence of moments was obtained, by other methods, in Reference3Reference4.

## 3. Correlation functions in the bulk of the spectrum

### 3.1. Proof of Theorem 3

We refer the reader to Section 1.3 of the Introduction for the definition of a regular sequence and the statement of Theorem 3. Also, in this section, we shall be working in the bulk of the spectrum, that is, we shall assume that all numbers defined in Equation1.10 lie inside The edges . of the spectrum and its exterior will be treated in the next section.

In our proof, we shall follow the strategy explained in Section 1.5. Namely, in order to compute the limit of we shall use the contour integral

compute the asymptotics of for and estimate , away from Both tasks will be accomplished using classical results about the Bessel functions. .

We start our proof with the following lemma which formalizes the above informal depoissonization argument. The hypothesis of this lemma is very far from optimal, but it is sufficient for our purposes. For the rest of this section, we fix a number which shall play an auxiliary role.

#### Lemma 3.1

Let be a sequence of entire functions

and suppose that there exist constants and such that

as Then .

#### Proof.

By replacing by we may assume that , By Cauchy and Stirling formulas, we have .

Choose some large and split the circle into two parts as follows:

The inequality Equation3.1 and the equality

imply that the integral decays exponentially provided is large enough. On the inequality ,Equation3.2 applies for sufficiently large and gives

Therefore, the integral is of the following integral:

Hence, and the lemma follows.

#### Definition 3.2

Denote by the algebra (with respect to term-wise addition and multiplication) of sequences which satisfy the properties Equation3.1 and Equation3.2 for some, depending on the sequence, constants and Introduce the map .

which is clearly a homomorphism.

#### Remark 3.3

Note that we do not require to be entire. Indeed, the kernel may have a square root branching; see the formula Equation2.14.

By Theorem 2, the correlation functions belong to the algebra generated by sequences of the form

where the sequence is regular which, we recall, means that the limits

exist, finite or infinite. Therefore, we first consider such sequences.

#### Proposition 3.4

If is regular, then

In the proof of this proposition it will be convenient to allow For complex sequences . we shall require the number ; may be arbitrary.

#### Lemma 3.5

Suppose that a sequence is as above and, additionally, suppose that , are bounded and Then the sequence . satisfies Equation3.2 with and certain .

#### Proof of Lemma 3.5.

We shall use Debye’s asymptotic formulas for Bessel functions of complex order and large complex argument; see, for example, Section 8.6 in Reference43. Introduce the function

The formula Equation1.9 can be rewritten as

The asymptotic formulas for Bessel functions imply that

where

provided that in such a way that stays in some neighborhood of the precise form of this neighborhood can be seen in Figure 22 in Section 8.61 of ;Reference43. Because we assume that

and because the ratios , , stay close to For future reference, we also point out that the constant in . in Equation3.4 is uniform in provided is bounded away from the endpoints .

First we estimate The function . clearly takes real values on the real line. From the obvious estimate

and the boundedness of , and , we obtain an estimate of the form

If then because of the denominator in ,Equation3.3 the estimate Equation3.5 implies that

Since it follows that in this case the lemma is established. ,

Assume, therefore, that is finite. Observe that for any bounded increment we have

and, in particular, the last term is Using the trigonometric identity .

and observing that

we compute

Since, by hypothesis,

and the lemma follows. ,

#### Remark 3.6

Below we shall need this lemma for a variable sequence Therefore, let us spell out explicitly under what conditions on . the estimates in Lemma 3.5 remain uniform. We need the sequences and to converge uniformly; then, in particular, the ratios and are uniformly bounded away from Also, we need . and to be uniformly bounded. Finally, we need to be uniformly bounded from below.

#### Proof of Proposition 3.4.

First, we check the condition Equation3.2. In the case this was done in the previous lemma. Suppose, therefore, that is a regular sequence in and consider the asymptotics of .

Because the function is an entire function of and we have

where is arbitrary; we shall take to be some small but fixed number. From the previous lemma we know that

From the above remark it follows that this estimate is uniform in This implies the property .Equation3.2 for .

To prove the estimate Equation3.1 we use Schläfli’s integral representation (see Section 6.21 in Reference43)

which is valid for and even for provided or .

If then the second summand in ,Equation3.8 vanishes and the first summand is uniformly in This implies the estimate .Equation3.1 provided .

It remains, therefore, to check Equation3.1 for where is a regular sequence. Again, we use Equation3.7. Observe that since the second summand in ,Equation3.8 is uniformly small provided is bounded from above and is bounded from below. Therefore, Equation3.7 produces the Equation3.1 estimate for For . we use the relation Equation2.13 and the reccurence Equation2.16 to obtain the estimate.

#### Proof of Theorem 3.

Let be a regular sequence and let the numbers and be defined by Equation1.10, Equation1.11. We shall assume that for all The validity of the theorem in the case when . for some will be obvious from the results of the next section.

We have

where the first line is the definition of and the second is Theorem 2. From Equation3.9 it is obvious that is entire. Therefore, we can apply Lemma 3.1 to it. It is clear that Lemma 3.1, together with Proposition 3.4, implies Theorem 3. The factorization Equation1.12 follows from the vanishing