aa r X i v : . [ m a t h . C A ] J a n HIRSCHMAN–WIDDER DENSITIES
ALEXANDER BELTON, DOMINIQUE GUILLOT, APOORVA KHARE, AND MIHAI PUTINAR
Abstract.
Hirschman and Widder introduced a class of P´olya frequency functionsgiven by linear combinations of one-sided exponential functions. The members of thisclass are probability densities, and the class is closed under convolution but not underpointwise multiplication. We show that, generically, a polynomial function of sucha density is a P´olya frequency function only if the polynomial is a homothety, andalso identify a subclass for which each positive-integer power is a P´olya frequencyfunction. We further demonstrate connections between the Maclaurin coefficients,the moments of these densities, and the recovery of the density from finitely manymoments, via Schur polynomials. Introduction and main results
The class of P´olya frequency functions is central to the theory of total positivity. Itsbasic properties were announced by Schoenberg in 1947–48 [18, 19] and further detailswere provided in subsequent work [20, 21]. These functions have been actively studiedever since.
Definition 1.1 (Schoenberg) . A function Λ : R → [0 , ∞ ) is a P´olya frequency function if it is Lebesgue integrable and non-zero at two or more points, and the Toeplitz kernel T Λ : R × R → R ; ( x, y ) Λ( x − y )is totally non-negative. This last statement means that, for any integer p ≥ x < · · · < x p and y < · · · < y p , the matrix (cid:0) Λ( x j − y k ) (cid:1) pj,k =1 has non-negative determinant.Schoenberg showed in [21] that the bilateral Laplace transform B{ Λ } ( s ) := Z R e − xs Λ( x ) d x of a P´olya frequency function Λ converges in a open strip containing the imaginary axis,and equals on this strip the reciprocal of an entire function Ψ in the Laguerre–P´olyaclass [10, 16], with Ψ(0) = 1. Conversely, any function Ψ of this form agrees with the re-ciprocal of the bilateral Laplace transform of some P´olya frequency function on its stripof convergence. Schoenberg also proved that a P´olya frequency function necessarily has Date : 6th January 2021.2010
Mathematics Subject Classification.
Key words and phrases.
P´olya frequency function, totally positive function, hypoexponentialdistribution. unbounded support, and either vanishes nowhere or vanishes on a semi-axis. Membersof the latter class of functions are said to be one sided , and Schoenberg [21] also char-acterized this subclass via the bilateral Laplace transform. This characterization alsoallows the non-smooth members of this subclass to be identified.
Theorem 1.2 (Schoenberg) . If the one-sided P´olya frequency function Λ vanisheson ( −∞ , then / B{ Λ } is the restriction of an entire function Ψ in the first Laguerre–P´olya class, so has the form Ψ( s ) = Ce δs ∞ Y j =1 (1 + α j s ) , where C > , δ ≥ , α j ≥ and < ∞ X j =1 α j < ∞ . (1.1) Conversely, if the entire function Ψ has the form (1.1) then there exists a P´olyafrequency function Λ that vanishes on ( −∞ , such that Ψ( s ) = 1 / B{ Λ } ( s ) on an openstrip containing the origin.Such a P´olya frequency function Λ is continuous and positive on ( δ, ∞ ) and vanisheson ( −∞ , δ ) . Furthermore, the function Λ is smooth if and only if α j is non-zero forinfinitely many j , and is continuous unless α j is non-zero for exactly one j . In this note, we study the one-sided P´olya frequency functions which are continuousbut non-smooth, that is, those with at least two but only finitely many non-zero terms α , . . . , α m in (1.1), and their powers. We may normalize so that Λ is a probabilitydensity function, whence C = 1, and we may also assume δ = 0, by replacing Λ with x Λ( x + δ ). Thus, we study the collection of finitely determined P´olya frequencyfunctions of the form Λ α , such that B{ Λ α } ( s ) = m Y j =1 (1 + α j s ) − , where α := ( α , . . . , α m ) and m ≥ . Some historical comments are appropriate, and here we recount this subject’s earlydevelopments in chronological order. In 1947, Schoenberg [18] announced the notion ofa P´olya frequency function. In their 1949 work, Hirschman and Widder [7] studied Λ α for distinct positive α , . . . , α m and its degree of smoothness, via the Laplace transform.This was followed by Schoenberg’s first full paper on P´olya frequency functions [21] in1951. In this work, Schoenberg placed the analysis of Hirschman and Widder in a widercontext, with the last part of Theorem 1.2 showing that the collection of Hirschman–Widder functions is dense, in a suitable sense, in the set of non-smooth one-sided P´olyafrequency functions. Finally, Hirschman and Widder’s 1955 monograph [8] contains adetailed analysis of these functions and their Laplace transforms, and provides ampleevidence for the relevance of such functions to operational calculus and approximationtheory.For this reason, we adopt the following terminology for this family of functions, nowallowing for non-distinct and negative α j . For brevity, we let R × := R \ { } denote theset of non-zero real numbers. Definition 1.3.
Given α = ( α , . . . , α m ) ∈ ( R × ) m , where m ≥
2, the corresponding
Hirschman–Widder density is the unique continuous function Λ α : R → [0 , ∞ ) with IRSCHMAN–WIDDER DENSITIES 3 bilateral Laplace transform Z R e − xs Λ α ( x ) d x = m Y j =1 (1 + α j s ) − (1.2)on the open half-plane { s ∈ C : ℜ s > − α − j for j = 1 , . . . , m } .The next section focuses on some basic properties of these functions.(1) Each such function Λ α exists and is unique.(2) The function Λ α is both a P´olya frequency function and a probability densityfunction. It is one sided if and only if all the entries of α have the same sign.(3) The function Λ α has a multiplicative representation via convolution, as well asan additive one involving one-sided exponentials.However, this collection of densities is not closed under pointwise multiplication. Theprincipal contribution of the present work is to identify when some simple algebraicoperations preserve the class of Hirschman–Widder densities and when they do not.More specifically, we focus on polynomial functions and study the generic behaviour ofthese operations and their departure from mapping the class of densities to itself. Themain result is as follows. Theorem 1.4. (1)
Suppose m ≥ . There exists a subset N of (0 , ∞ ) m with Lebesgue measurezero such that p ◦ Λ α : x p (cid:0) Λ α ( x ) (cid:1) is not a P´olya frequency function for any α ∈ (0 , ∞ ) m \ N and any real poly-nomial p that is not a homothety, that is, p ( x ) cx for any c > . (2) Suppose α = ( α , . . . , α m ) ∈ (0 , ∞ ) m , where m ≥ , is such that the reciprocals a := α − , . . . , a m := α − m form an arithmetic progression. Then c Λ n α is aP´olya frequency function for every c > and every integer n ≥ . If, moreover, α = α m or α /α is irrational, then p ◦ Λ α is not a P´olya frequency functionfor any other real polynomial p . We remark that the first assertion with m = 3 and p ( x ) = x n was shown in our recentpreprint [3], and played a key role in characterizing those post-composition transformsthat preserve the class of one-sided P´olya frequency functions. Theorem 1.4 shows thatthese m = 3 examples are merely the first in a large, multi-parameter family of P´olyafrequency functions with the same property.In the case where p ( x ) = x n , note that the assumption m ≥ m = 2 case is covered by the second assertion, since everypair of numbers is trivially in arithmetic progression.As is discussed in Section 2.4, the Hirschman–Widder densities are connected toclassical probability theory. In the present work, we connect them to another area: thetheory of symmetric functions, and specifically Schur polynomials. In fact, this also hasa connection to probability: we show below that these densities can be reconstructedfrom finitely many moments, via symmetric function identities. ALEXANDER BELTON, DOMINIQUE GUILLOT, APOORVA KHARE, AND MIHAI PUTINAR
Definition 1.5.
Given a field F of size at least m ≥
2, and a tuple λ = ( λ , . . . , λ m )of non-negative integers λ ≤ · · · ≤ λ m , we define the corresponding Schur polynomial to be the polynomial extension of the function s λ ( a , . . . , a m ) := det( a λ k j ) mj,k =1 V ( a , . . . , a m )for distinct a , . . . , a m ∈ F , where V ( a , . . . , a m ) := det( a k − j ) = Q ≤ j Given α = ( α , . . . , α m ) ∈ (0 , ∞ ) m , where m ≥ , the Hirschman–Widder density Λ α is represented by its Maclaurin series on [0 , ∞ ) , with n th coefficient Λ ( n ) α (0 + ) = ( if ≤ n ≤ m − , ( − n − m +1 α − · · · α − m s (0 , ,...,m − ,n ) ( α − , . . . , α − m ) if n ≥ m − . The density Λ α has p th moment µ p := Z R x p Λ α ( x ) d x = p ! s (0 , ,...,m − ,m − p ) ( α , . . . , α m ) if p ≥ . The parameter α can be recovered, up to permutation of its entries, from the first m moments, µ , . . . , µ m , and also from the first m + 1 non-trivial Maclaurin coefficients, Λ ( m − α (0 + ) , . . . , Λ (2 m − α (0 + ) . To the best of our knowledge, these connections between the moments, the Maclaurincoefficients and the moment-recovery problem for Hirschman–Widder densities have notpreviously been noted in the literature. While the moments are computable from firstprinciples using probability theory, we provide another recipe: the moment-generatingfunction may be obtained by evaluating the generating function of the complete homo-geneous symmetric polynomials. In a similar spirit, the moment-recovery problem isseen to be intimately connected with the Jacobi–Trudi identity. IRSCHMAN–WIDDER DENSITIES 5 Acknowledgements. D.G. was partially supported by a University of DelawareResearch Foundation grant, by a Simons Foundation collaboration grant for mathe-maticians, and by a University of Delaware strategic initiative grant.A.K. was partially supported by Ramanujan Fellowship grant SB/S2/RJN-121/2017,MATRICS grant MTR/2017/000295, SwarnaJayanti Fellowship grants SB/SJF/2019-20/14 and DST/SJF/MS/2019/3 from SERB and DST (Govt. of India), and by grantF.510/25/CAS-II/2018(SAP-I) from UGC (Govt. of India).M.P. was partially supported by a Simons Foundation collaboration grant.2. Hirschman–Widder densities and symmetric functions In this section, we prove Theorem 1.6. We begin by recalling two approaches toconstructing Hirschman–Widder densities, guided by the original memoir [8].2.1. Two constructions of Hirschman–Widder densities. Following Hirschmanand Widder [8], we first establish the existence of their eponymous densities via theconvolution product. Proposition 2.1. Let α = ( α , . . . , α m ) ∈ ( R × ) m , where m ≥ . (1) The corresponding Hirschman–Widder density Λ α exists and is unique. (2) The function Λ α is both a P´olya frequency function and a probability densityfunction. (3) The function Λ α is one sided if and only if the entries of α all have the samesign.Proof. If α > ϕ α : R → [0 , ∞ ); x ( x < ,α − e − α − x if x ≥ α < ϕ α : x ϕ − α ( − x ) is also aP´olya frequency function. The functionΛ α := ϕ α ∗ · · · ∗ ϕ α m (2.1)is continuous, being a convolution product, and its bilateral Laplace transform is asrequired, since B{ ϕ α } ( s ) = (1 + αs ) − .The class of P´olya frequency functions is closed under convolution [21, Lemma 5], andso Λ α is a P´olya frequency function. It is a probability density because B{ Λ α } (0) = 1.For uniqueness, we note that any continuous function with prescribed bilateralLaplace transform F such that t F (i t ) is integrable can be recovered everywherevia the Fourier–Mellin integral: here,Λ α ( x ) = 12 π i Z +i ∞− i ∞ e xs (1 + α s ) · · · (1 + α m s ) d s ( x ∈ R ) . (2.2)If α j and α k have opposite signs then a short calculation shows that ( ϕ α j ∗ ϕ α k )( x ) > x ∈ R . Furthermore, if f is continuous and positive on R then ϕ α ∗ f is alsocontinuous and positive on R , for any α ∈ R × . This gives one part of (3) and theconverse is immediate. (cid:3) ALEXANDER BELTON, DOMINIQUE GUILLOT, APOORVA KHARE, AND MIHAI PUTINAR As the Laplace transform does not behave well for the product given by pointwisemultiplication, it is useful to have a second construction for the function Λ α .Given one-sided exponentials x ≥ e − a x , . . . , x ≥ e − a k x , where k ≥ < a < · · · < a k , there are, up to homothety, only finitely many choices of real coefficients c , . . . , c k such that the linear combination x ≥ P kj =1 c j e − a j x is a P´olya frequency function. Moregenerally, we have the following result, where each coefficient c j may be a polynomial. Definition 2.2. For a tuple of positive real numbers a = ( a , . . . , a k ) such that k ≥ a < · · · < a k , and a tuple of real polynomials c = ( c , . . . , c k ), letΛ a , c : R → R ; x (P kj =1 c j ( x ) e − a j x if x ≥ , x < . We let deg p denote the degree of the polynomial p , with deg p := −∞ if p = 0. Proposition 2.3. Suppose a = ( a , . . . , a k ) is a tuple of positive real numbers suchthat k ≥ and a < · · · < a k . (1) Given any tuple of non-negative integers n = ( n , . . . , n k ) , not all zero, thereexists a unique tuple c = c a , n of real polynomials such that B{ Λ a , c } ( s ) = k Y j =1 (1 + α j s ) − n j , where α j := a − j for j = 1 , . . . , k. (2.3) The tuple c a , n is such that deg c a , n j ≤ n j − for all j .In particular, if α = ( α , . . . , α m ) ∈ (0 , ∞ ) m , a = ( a , . . . , a k ) ∈ (0 , ∞ ) k and m = ( m , . . . , m k ) are such that a < · · · < a k and a − j appears exactly m j times in α , with m + · · · + m k = m ≥ , then the function Λ a , c with c = c a , m is the Hirschman–Widder density Λ α . (2) If the tuple of real polynomials c is not of the form t c a , n for some t > and somenon-zero tuple of non-negative integers n , then Λ a , c is not a P´olya frequencyfunction.Proof. For (1), partial-fraction decomposition gives a family of real coefficients c a , n such that k Y j =1 (1 + α j s ) − n j = k X j =1 n j X l =1 c a , n j,l (1 + α j s ) − l . Furthermore, if a > α := a − then B{ x ≥ x l − e − ax } ( s ) = ( l − α l (1 + αs ) − l ( l = 1 , , . . . ) . It follows that setting c a , n j ( x ) = n j X l =1 a lj c a , n j,l ( l − x l − ( j = 1 , . . . k ) IRSCHMAN–WIDDER DENSITIES 7 gives the existence of c a , n as required. Uniqueness follows by Lerch’s theorem: if B{ Λ a , c } ≡ B{ Λ a , c ′ } on some half plane then Λ a , c = Λ a , c ′ , as both restrict to continuousfunctions on [0 , ∞ ) and vanish elsewhere. Hence it suffices to show that k X j =1 p j ( x ) e − a j x ≡ ⇒ p ( x ) = · · · = p k ( x ) ≡ p , . . . , p k , but this follows because k X j =1 p j ( x ) e − a j x ≡ ⇒ p ( x ) + k X j =2 p j ( x ) e ( a − a j ) x ≡ ⇒ lim x →∞ p ( x ) = 0 , whence p = 0, and so on. For the final claim, we need only show that Λ a , c is continuousat the origin, that is, 0 = k X j =1 c j (0) = X j : m j > a j c a , m j, , but this holds because k X j : m j > a j c a , m j, = lim s →∞ k X j =1 m j X l =1 c a , m j,l s (1 + α j s ) − l = lim s →∞ s k Y j =1 (1 + α j s ) − m j = 0 . For (2), suppose c is not of the form t c a , n as asserted. Then B{ Λ a , c } ( s ) is a rationalfunction of the form p ( s ) / Q kj =1 (1 + α j s ) m j , with the numerator polynomial p not afactor of the denominator. Hence the reciprocal of B{ Λ a , c } is not the restriction of afunction belonging to the Laguerre–P´olya class. By Theorem 1.2, Λ a , c is not a P´olyafrequency function. (cid:3) Vandermonde matrices and closed-form expressions. In the first part ofTheorem 1.4, the generic tuple α will be taken to have distinct entries. We mayassume without loss of generality that these are strictly decreasing, whence the entriesof the corresponding reciprocal tuple a are strictly increasing. The next result providesa closed-form expression for the unique tuple c such that Λ α = Λ a , c is a Hirschman–Widder density. Note first that Proposition 2.3 implies that c consists of polynomialsof degree zero, that is, constants. Proposition 2.4. Let α ∈ (0 , ∞ ) m be such that m ≥ and α > · · · > α m , and let a = ( a , . . . , a m ) , where a j := α − j for j = 1 , . . . , m . The Hirschman–Widder density Λ α = Λ a , c , where c = ( c , . . . , c m ) ∈ ( R × ) m is such that c j = a j Y k = j a k a k − a j ( j = 1 , . . . , m ) . In particular, the constants c , . . . , c m alternate in sign and sum to zero. Here we provide a different proof to that of Hirschman and Widder [8, Section X.2.2]and so demonstrate a connection between these densities and the theory of symmetricfunctions. We employ an algebraic lemma involving alternating polynomials.In the following definition and lemma, we let F denote an arbitrary field. ALEXANDER BELTON, DOMINIQUE GUILLOT, APOORVA KHARE, AND MIHAI PUTINAR Definition 2.5. Given any a := ( a , . . . , a m ) ∈ F m , where m ≥ 2, let b a j ∈ F m − beobtained by removing the j th term from a , so that b a j := ( a , . . . , a j − , a j +1 , . . . , a m ) ∈ F m − ( j ∈ , . . . , m ) . Recall that V ( a ) := Q ≤ j Given any a ∈ F m , with m ≥ , the following identity holds in thepolynomial ring F [ X ] : V ( a ) X l = m X j =1 ( − j + l − a lj V ( b a j ) Y k = j ( X + a k ) ( l = 0 , . . . , m − . Proof. Suppose first that not all the entries of a are distinct, say a p = a q for p = q .Then V ( a ) vanishes, as do the summands on the right-hand side for j not equal to p or q , whereas the remaining two summands cancel each other, since b a q can be obtainedfrom b a p by | p − q | − a has distinct entries. Since both sides are polynomials in X ofdegree at most m − 1, it suffices to show they agree at − a p for p = 1, . . . , m . However,when evaluated at − a p , the right-hand side reduces to( − a p ) l V ( b a p ) Y k p ( a k − a p ) , which is precisely V ( a )( − a p ) l . (cid:3) Remark 2.7. Lemma 2.6 can be applied to compute the inverse of the matrix E ( a ) := e ( b a ) e ( b a ) · · · e ( b a m ) e ( b a ) e ( b a ) · · · e ( b a m )... ... . . . ... e m − ( b a ) e m − ( b a ) · · · e m − ( b a m ) , (2.4)where a = ( a , . . . , a m ) contains distinct elements of the field F and e l is the elementarysymmetric polynomial e ≡ e l ( b , . . . , b n ) := X ≤ j The matrix identity (2.6) provides a closed-form expression for the inverse of a standardVandermonde matrix over any field. This formula can also be deduced from an alternateexpression for the inverse; see [13].A further consequence of (2.6) is the following expression for the determinant of E ( a ):det E ( a ) = ( − m ( m − / V ( a ) . (2.7)This formula was proved by a different method in [12, pp.41–42].With Lemma 2.6 at hand, we now obtain the aforementioned closed-form expressionfor the Hirschman–Widder density. Proof of Proposition 2.4. Let α and a be as in the statement of the Proposition, anddefine c by letting c j := a j Y k = j a k a k − a j = ( − j − V ( b a j ) V ( a ) m Y k =1 a k ( j = 1 , . . . , m ) . (2.8)The bilateral Laplace transform B{ Λ a , c } ( s ) = m X j =1 ( − j − V ( b a j ) V ( a )( s + a j ) m Y k =1 a k = a · · · a m ( s + a ) · · · ( s + a m ) = B{ Λ α } ( s ) , by Lemma 2.6 with l = 0 and X = s . We therefore conclude that Λ α = Λ a , c , byProposition 2.3(1). That c , . . . , c m are alternating follows because c j has the same signas ( − j − , and that they sum to zero was shown in the proof of Proposition 2.3(1). (cid:3) Remark 2.8. The explicit form of Λ a , c obtained above provides a connection to thetopic of cardinal L -splines ; see [14], for example, for more on these. In [22, Section 5],the authors study the restriction of a certain L -spline to an interval [0 , η ]. With a slightchange of notation to match that used here, this is e A m ( − x ; t ) = 1 + m X j =1 e − a j x (1 − t ) t − e a j η Y k = j a k a k − a j . It follows immediately from Proposition 2.4 that the Hirschman–Widder density Λ a , c is the asymptote of this spline:Λ a , c ( x ) = lim t →±∞ dd x e A m ( − x ; t ) . Non-smoothness at the origin. We now take a closer look at Schoenberg’sresult [21, Corollary 2] that Hirschman–Widder densities are the only non-smooth,continuous one-sided P´olya frequency functions, up to homothety. In the process,we obtain what is, to the best of our knowledge, a novel connection between P´olyafrequency functions and symmetric functions.The theme in this subsection is the smoothness of the map α Λ α ( x ), where x isfixed; for simplicity, we consider only α with positive entries. The simplest example is such thatΛ ( α ,α ) ( x ) = ( α − α ) − ( e − α − x − e − α − x ) if α = α and x > ,α − xe − α − x if α = α and x > , . It is readily verified that the map ( α , α ) Λ ( α ,α ) ( x ) is continuous on (0 , ∞ ) forany x ∈ R . Indeed, more is true.(1) For any non-negative integer l and any x ∈ R × , the map(0 , ∞ ) → R ; ( α , α ) Λ ( l )( α ,α ) ( x )is real analytic.(2) The left and right limits at 0 of the first derivative of Λ ( α ,α ) are distinct:Λ ′ ( α ,α ) (0 − ) := lim x → − Λ ′ ( α ,α ) ( x ) = 0and Λ ′ ( α ,α ) (0 + ) := lim x → + Λ ′ ( α ,α ) ( x ) = α − α − . In particular, the Hirschman–Widder density Λ ( α ,α ) is not twice differentiableat the origin.The following result is a generalization of the claim (1). Proposition 2.9. For any non-negative integer l and any x ∈ R × , the map α Λ ( l ) α ( x ) is real analytic on (0 , ∞ ) m , for any m ≥ .Proof. Let α = ( α , . . . , α m ) ∈ (0 , ∞ ) m , where m ≥ 2, and recall the identity (2.2):Λ α ( x ) = 12 π Z ∞−∞ e i tx m Y j =1 (1 + i α j t ) − d t ( x ∈ R ) . There is no obstruction to extending this integral to the case where α , . . . , α m lie inthe open right half-plane H + := { z ∈ C : ℜ z > } . Indeed, for such α , . . . , α m , wehave that | e i tx m Y j =1 (1 + i α j t ) − | ≤ | t | − m m Y j =1 |ℜ α j | − , which is Lebesgue integrable with respect to t . The analyticity of the integrand in thecomplex variables α , . . . , α m is then inherited by the integral.We fix x ∈ R × . Integration by parts yields the identity( − i x ) n Λ α ( x ) = 12 π Z ∞−∞ e i tx G n ( t ) d t for any non-negative integer n , where G n ( t ) := ∂ n ∂t n (cid:16) m Y j =1 (1 + i α j t ) − (cid:17) = O ( | t | − n − m ) as | t | → ∞ , since G n ( t ) is a homogeneous polynomial in (1 + i α t ) − , . . . , (1 + i α m t ) − of degree n + m . IRSCHMAN–WIDDER DENSITIES 11 Now suppose n ≥ l − m + 2, and note that the derivative F l,n, α ( x ) := ∂ l ∂x l (cid:16) ( − i x ) n Λ α ( x ) (cid:17) = 12 π Z ∞−∞ (i t ) l e i tx G n ( t ) d t. Differentiation under the integral sign is valid here as long as the function t 7→ | t | l | G n ( t ) | is integrable, and this holds because l − n − m ≤ − 2. The integrand is an analyticfunction in the variables α , . . . , α m as long as these all lie in H + , and an inductiveargument now gives the result, as Λ ( l ) α ( x ) can be expressed as a linear combinationof F l,n, α ( x ) and derivatives of lower order. (cid:3) We now examine the right-hand derivatives at the origin of our Hirschman–Widderdensities; all of the left-hand derivatives here are clearly zero. Our approach will involveanother instance of Schur polynomials.We have already seen that the elementary symmetric polynomials of Remark 2.7are Schur polynomials. Another well studied family of symmetric functions is thatof complete homogeneous symmetric polynomials: if l is a non-negative integer and a = ( a , . . . , a m ) ∈ F m for some m ≥ h ≡ h l ( a ) := X ≤ j ≤ j ≤···≤ j l ≤ m a j a j · · · a j l . (2.9)These are also Schur polynomials: by [12, (3.9)], we have that h l = s λ , where λ = (0 , , . . . , m − , m − l ) . (2.10)Having defined these polynomials, we may now state and prove the following result. Proposition 2.10. Suppose that a = ( a , . . . , a m ) ∈ (0 , ∞ ) m , where m ≥ , andlet α := ( α = a − , . . . , α m = a − m ) . The Hirschman–Widder density Λ α has theMaclaurin-series expansion Λ α ( x ) = a · · · a m ∞ X n = m − ( − n − m +1 h n − m +1 ( a , . . . , a m ) n ! x n (2.11) valid for all x ∈ [0 , ∞ ) . Consequently, the function Λ α is m − times continuouslydifferentiable, but Λ ( m − α (0) does not exist.Proof. We assume first that a , . . . , a m are distinct and, without loss of generality,that a < · · · < a m . If n is a non-negative integer then, using the explicit form of c j from (2.8), we see thatΛ ( n ) α (0 + ) = a · · · a m V ( a ) m X j =1 ( − j − V ( b a j )( − a j ) n = ( − n a · · · a m V ( a ) det a n a n · · · a nm · · · a a · · · a m a a · · · a m ... ... . . . ... a m − a m − · · · a m − m . If n = 0, 1, . . . , m − 2, then the matrix above has two identical rows, so its determinantvanishes. For n ≥ m − 1, moving the top row to the bottom takes m − α is the restriction to [0 , ∞ ) of an entire function.Since α Λ α ( x ) is real analytic on (0 , ∞ ) m for any x > 0, by Proposition 2.9, itis continuous there. As the right-hand side of (2.11) is also a continuous function ofsuch α for fixed x > 0, the identity holds for all ( x, α ) ∈ (0 , ∞ ) m +1 . It also holdstrivially when x = 0.An obstruction to Λ α being continuously differentiable can only appear at the origin,where Λ ( n ) α (0 − ) is always zero. The working above shows that Λ ( n ) α (0 + ) is also zero if n = 0, . . . , m − 2, whereas Λ ( m − α (0 + ) = h ( a , . . . , a m ) = 1. (cid:3) As a brief digression, we use Proposition 2.10 to obtain a classical identity in algebraiccombinatorics. Although the identity is well known, its connection to P´olya frequencyfunctions is not.We begin with the following elementary observation. Lemma 2.11. Let a = ( a , . . . , a m ) ∈ [0 , ∞ ) m , where m ≥ . Then lim l →∞ h l ( a ) /l = max { a , . . . , a m } . Proof. If a k ≥ a j for j = 1, . . . , m and l ≥ m − a lk ≤ h l ( a , . . . , a m ) ≤ (cid:18) l + m − l (cid:19) a lk ≤ (2 l ) m − a lk ( m − . The result follows. (cid:3) Corollary 2.12. Let a = ( a , . . . , a m ) ∈ (0 , ∞ ) m , where m ≥ . The generatingfunction of the family of complete homogeneous symmetric polynomials in a , . . . , a m is ∞ X l =0 h l ( a , . . . , a m ) z l = m Y j =1 − a j z whenever | z | < min { a − j : j = 1 , . . . , m } . As we explain below, this corollary provides a way to obtain the moments of Λ α from its Maclaurin coefficients. The reverse inference can be drawn if one knows themoments, via arguments of P´olya [17] and Schoenberg [21]. This is discussed furtherin Remark 2.18 below. Proof. This power series has radius of convergence min { a − j : j = 1 , . . . , m } , byLemma 2.11. Now suppose 0 > z > max {− a − j : j = 1 , . . . , m } and let s := − z − .Setting α j := a − j for j = 1, . . . , m , we see that m Y j =1 − a j z = α · · · α m s m B{ Λ α } ( s )= α · · · α m s m Z ∞ e − sx Λ α ( x ) d x IRSCHMAN–WIDDER DENSITIES 13 = s m Z ∞ ∞ X n = m − ( − n − m +1 h n − m +1 ( a , . . . , a m ) n ! x n e − sx d x, by Proposition 2.10. As the series s m ∞ X n = m − h n − m +1 ( a , . . . , a m ) n ! Z ∞ x n e − sx d x = ∞ X l =0 h l ( a , . . . , a m ) s − l is absolutely convergent, we may exchange the order of integration and summation inthe previous formula to see that the product Q mj =1 (1 − a j z ) − equals ∞ X l =0 h l ( a , . . . , a m ) z l , as claimed. We conclude by the identity theorem. (cid:3) Hirschman–Widder densities and probability theory. We now explore someconnections to probability theory. A natural first question is to identify the randomvariables distributed with Hirschman–Widder density functions. Proposition 2.13. Let α ∈ (0 , ∞ ) m , where m ≥ . Then the Hirschman–Widderdensity Λ α is the probability density function for the random variable α X + · · · + α m X m , where X , . . . , X m are independent and identically distributed exponential randomvariables with mean .Proof. A exponential random variable X with mean 1 has density function x ≥ e − x ,so if α > αX has density function x ≥ α − e − α − x = ϕ α ( x ). The result nowfollows by (2.1). (cid:3) Remark 2.14. Hirschman–Widder density functions are studied in the probability andstatistics literature under the name of hypoexponential densities . They are intimatelyconnected to the time to absorption for a finite-state Markov chain. When the entriesof α are equal, then Λ α is an Erlang density , named after the father of queueing theory;this is a special case of the gamma distribution, occurring when the shape parameter isan integer. These densities have found use in diverse applied fields, including queueingtheory, population genetics, reliability analysis and cell biology. The connection toHirschman and Widder’s memoir seems to be generally unnoticed in the probabilityliterature.The probabilistic interpretation also leads to closed-form expressions. The explicitformula for Λ α ( x ) when α , . . . , α m are positive and distinct, which appears inHirschman and Widder’s memoir in analysis [8], also appears in probability textbooks;see, for example, Exercise 12 in Chapter I of Feller’s 1966 book [5]. An explicit formulain the case where repeats may occur was obtained by Amari and Misra [2]. Remark 2.15. Schoenberg’s characterization of P´olya frequency functions, as thosefor which the reciprocal of the bilateral Laplace transform is the restriction of an entirefunction in the Laguerre–P´olya class, admits the same probabilistic interpretation. The Hadamard–Weierstrass factorization implies that such a function is the density functionof a possibly infinite linear combination of independent and identically distributedexponential random variables, together with at most one Gaussian random variable.We now obtain the promised closed form for the moments of the random variablesdistributed according to Hirschman–Widder densities. Proof of Theorem 1.6. The first part of this result has been established in the proof ofProposition 2.10. For the second, suppose first that the entries of α = ( α , . . . , α m ) aredistinct, and let a j := α − j for j = 1, . . . , m . Suppose further without loss of generalitythat a < · · · < a m . If p is a non-negative integer then (2.8) gives that1 p ! Z ∞−∞ x p Λ α ( x ) d x = a · · · a m p ! V ( a ) m X j =1 ( − j − V ( b a j ) Z ∞ x p e − a j x d x = a · · · a m V ( a ) m X j =1 ( − j − V ( b a j ) a − p − j = a · · · a m V ( a ) det a − p − a − p − · · · a − p − m · · · a a · · · a m ... ... . . . ... a m − a m − · · · a m − m . This quantity is unchanged if, for j = 1, . . . , m , we multiply the j th column by α m − j ,and multiply the whole expression by ( a · · · a m ) m − , to obtain( a · · · a m ) m − V ( a ) det α m − p α m − p · · · α m − pm α m − α m − · · · α m − m α m − α m − · · · α m − m ... ... . . . ...1 1 · · · = ( a · · · a m ) m − V ( a ) ( − m ( m − / V ( α ) s (0 , ,...,m − ,m − p ) ( α , . . . , α m ) . As V ( a ) = ( a · · · a m ) m − ( − m ( m − / V ( α ) , this gives the result when α has distinct entries. The general case now follows by acontinuity argument.Finally, we explain how to recover the parameter α from moments or Maclaurincoefficients. The p th moment of Λ α is µ p := Z R x p Λ α ( x ) d x = p ! h p ( α ) ( p = 1 , , . . . ) . (2.12)The Jacobi–Trudi identity [12, (3.4)] asserts, for any increasing tuple ( λ , . . . , λ m ), that s ( λ ,...,λ m ) ( α ) = det (cid:0) h λ m − j +1 − m + k ( α ) (cid:1) mj,k =1 = det (cid:0) h λ j − k +1 ( α ) (cid:1) mj,k =1IRSCHMAN–WIDDER DENSITIES 15 with the conventions of Definition 1.5 and h l := 0 whenever l < 0. In particular, e l ( α ) = s (0 , ,...,m − l − ,m − l +1 ,...,m ) ( α ) = det (cid:18) A l C l D l (cid:19) = det D l ( l = 1 , . . . , m ) , where A l is a lower-triangular matrix with h ( α ) = 1 as each entry of the leadingdiagonal and D l is the l × l Toeplitz matrix h ( α ) 1 0 · · · h ( α ) h ( α ) 1 · · · h ( α ) h ( α ) h ( α ) · · · h l − ( α ) h l − ( α ) h l − ( α ) · · · h ( α ) 1 h l ( α ) h l − ( α ) h l − ( α ) · · · h ( α ) h ( α ) . It follows that e l ( α ) = det D l = f l ( µ , . . . , µ l ) for some polynomial function f l . Hence F ( t ) := 1 + m X l =1 f l ( µ , . . . , µ l ) t l = m X l =0 e l ( α ) t l = m Y j =1 (1 + α j t ) (2.13)is determined by the moments µ , . . . , µ m and the roots of F yield precisely the entriesof α .Similarly, Proposition 2.10 gives thatΛ ( n ) α (0 + ) / Λ ( m − α (0 + ) = ( − n − m +1 h n − m +1 ( α − , . . . , α − m ) ( n = m, . . . , m − α − , . . . , α − m ) from these ratios of Maclaurin coefficients. (cid:3) Remark 2.16. The computation of the moments of Λ α in the previous proof wasobtained with the assistance of the theory of symmetric functions. A more directapproach, using the probabilistic interpretation of Λ α , is also available. If X , . . . , X m are independent exponential random variables, each with mean one, and α , . . . , α m are positive constants, then the random variable X := P mj =1 α j X j has density Λ α , byProposition 2.13, and moment-generating function ∞ X p =0 µ p p ! z p = E [ e zX ] = B{ Λ α } ( − z ) = m Y j =1 − α j z . Corollary 2.12, with a replaced by α , now shows that the moments are as claimed.Alternately, we may proceed via an explicit computation: E (cid:2)(cid:16) m X j =1 α j X j (cid:17) p (cid:3) = p ! h p ( α , . . . , α m )for any non-negative integer p , since (cid:16) m X j =1 α j X j (cid:17) p = X p + ··· + p m = p (cid:18) pp · · · p m (cid:19) α p · · · α p m m X p · · · X p m m , where the sum is taken over non-negative integers, and E [ X p ] = p !. We now provide a connection between Hirschman–Widder densities and certain P´olyafrequency sequences. Karlin, Proschan, and Barlow proved that a probability density isa P´olya frequency function if and only if the sequence of normalized moments is a P´olyafrequency sequence, in that the corresponding Toeplitz matrix is totally non-negative;see [9, Theorem 3]. This Toeplitz matrix is formed from a bi-infinite extension of thenormalized-moments sequence, which here is . . . , , , , , h ( α ) , h ( α ) , h ( α ) , . . . and the total non-negativity of the corresponding Toeplitz matrix is again the numericalshadow of the Jacobi–Trudi identity.The observations above lead to the solution of the following moment problem. Corollary 2.17. Suppose µ = ( µ , . . . , µ m ) ∈ R m , where m ≥ , let µ := 1 and letthe l × l Toeplitz matrix D l := µ · · · µ / µ · · · µ / µ / µ · · · ... ... ... . . . ... ... µ l − / ( l − µ l − / ( l − µ l − / ( l − · · · µ µ l /l ! µ l − / ( l − µ l − / ( l − · · · µ / µ for l = 1 , . . . , m . The following are equivalent. (1) There exists α = ( α , . . . , α m ) ∈ (0 , ∞ ) m such that µ is the truncated momentsequence of the Hirschman–Widder density Λ α . (2) The generating polynomial F ( t ) := 1 + m X l =1 t l det D l of the determinant sequence (det D , . . . , det D m ) has all of its roots in ( −∞ , ,so is in the first Laguerre–P´olya class. (3) The sequence ( b n ) ∞ n = −∞ of the form . . . , , , , b = 1 , det D , . . . , det D m , , , , . . . is a P´olya frequency sequence, in that det( b p j − q k ) lj,k =1 is non-negative for anychoice of integers p < · · · < p l and q < · · · < q l , where l ≥ .Proof. That (1) = ⇒ (2) follows from the proof of Theorem 1.6. For the reverseimplication, it follows from (2) that there exists α ∈ (0 , ∞ ) m with e l ( α ) = det D l for l = 1, . . . , m . Expanding the determinant along its bottom row shows that µ l can berecovered from det D l and µ , . . . , µ l − , so Λ α has moments as required.The equivalence (2) ⇐⇒ (3) follows from immediately from [1, Theorem 6]. (cid:3) Remark 2.18. We conclude this section with a line of enquiry motivated by historicalconsiderations. P´olya showed in his 1915 paper [17] that a function Ψ in the Laguerre–P´olya class with the expansion1Ψ( s ) = µ − µ s + µ s − µ s + · · · (2.14) IRSCHMAN–WIDDER DENSITIES 17 and such that Ψ(0) > µ µ · · · µ n µ µ · · · µ n +1 ... ... . . . ... µ n µ n +1 · · · µ n > n = 0 , , , . . . . (2.15)In 1920, Hamburger went the other way [6] and deduced the existence of an underlyingdensity function Λ from the positivity of these determinants. This is precisely what ledSchoenberg to study these maps and to develop the theory of P´olya frequency functions.As observed by Schoenberg in [21], the coefficient µ p in (2.14) is precisely the p thmoment of the P´olya frequency function Λ. When Λ = Λ α , each normalized moment isa complete homogeneous symmetric polynomial in the entries of α , and so this Hankeldeterminant is also a symmetric polynomial. The Jacobi–Trudi identity shows thatthe Toeplitz moment determinant in the proof of Theorem 1.6 is monomial positive (apositive linear combination of monomials) and even Schur positive (a positive linearcombination of Schur polynomials). It is natural to ask if something similar appliesto the Hankel moment determinant, but it turns out that neither property holds ingeneral. For example, if m = 2 then µ p = p ! p X l =0 α l α p − l for p = 0 , , , . . . , so det (cid:18) µ µ µ µ (cid:19) = α + α , which is monomial positive but not Schur positive, anddet µ µ µ µ µ µ µ µ µ = 4 α + 12 α α − α α + 12 α α + 4 α , which is not even monomial positive.We have that Ψ( s ) = Q mj =1 (1 + α j s ), so the formula (2.12) for the moments and theexpansion (2.14) give another proof of Corollary 2.12: m Y j =1 (1 − α j s ) − = 1Ψ( − s ) = ∞ X l =0 µ l l ! s l = ∞ X l =0 h l ( α , . . . , α m ) s l . This line of thinking, together with (2.13), also provides the identity ∞ X l =0 µ l l ! s l = m Y j =1 (1 − α j s ) − = 1 F ( − s ) = 11 + P mk =1 f k ( µ , . . . , µ k )( − s ) k . Assuming, as is common in some applied areas, that moments are the only quantitiesavailable via measurements, we elaborate an alternate reconstruction scheme whichparallels well known algorithms used in inverse problems. More specifically, we note that ( µ j ) ∞ j =0 , as a Stieltjes moment sequence, is characterized by the positivity of theHankel determinants of the form (2.15) as well as those with a shifted index,det( µ j + k +1 ) nj,k =0 > n = 0 , , , . . . . For a proof of this, see, for instance [15, § P mk =1 f k ( µ , . . . , µ k )( − s ) k has only real and negative roots can beachieved by using its Sturm sequence.A classical result attributed to Kronecker asserts that the formal series ∞ X j =0 γ j s j represents a rational function if and only there exist positive integers n and r suchthat det( γ r + j + k ) nj,k =0 = 0 for all r ≥ r , and the minimum value of r is the degree of the denominator of the rational function.In principle, to verify this criterion involves an infinite sequence of vanishing Hankeldeterminants. Here, the hidden positivity in the moments of a Hirschman–Widderdistribution allows a drastic reduction of Kronecker’s criterion, to a single vanishingdeterminant.To this aim, and in order to identify the denominator Ψ( s ) = F ( s ) in the momentgenerating series without splitting it into factors, we appeal to the theory of cumulants.The cumulant series for Λ α is K ( s ) = ∞ X k =1 ν k s k := log ∞ X l =0 µ l l ! s l = − m X j =1 log(1 − α j s )and this is convergent whenever | s | < min { α − , . . . , α − m } . Its derivative has the seriesrepresentation K ′ ( s ) = ∞ X k =1 kν k s k − = m X j =1 α j − α j s = m X j =1 ( α j + α j s + α j s + · · · ) , so the k th cumulant ν k = 1 k m X j =1 α kj ( k = 1 , , , . . . ) . (2.16)As K ′ is the Stieltjes transform of finitely many point masses, the cumulant-generatingfunction admits the representation K ( s ) = Z ∞ log (cid:16) tt − s (cid:17) d σ ( t ) ( | s | < min { α − , . . . , α − m } ) , where σ is the sum of unit point masses at α − , α − , . . . , α − m . In other words,the cumulant-generating function K coincides, up to a constant and for small valuesof its argument, with the logarithmic potential of equally distributed masses at thereciprocals of the entries of α .As a final step in our reconstruction process, we apply the Pad´e approximationscheme to the series representing K ′ ( s ). We know from Kronecker’s criterion that IRSCHMAN–WIDDER DENSITIES 19 the minimal choice of n such that det (cid:0) ( k + j + 1) ν j + k +1 (cid:1) nj,k =0 = 0 is when n = m .Elementary matrix-algebra operations single out a unique pair of polynomials, a monicpolynomial P ( z ) with degree m and Q ( z ) with degree m − 1, such that P ( z ) m +1 X k =1 kν k z k = Q ( z ) + O (cid:16) z (cid:17) . Since K ′ ( s ) is the Stieltjes transform of a positive measure, we infer the equality offormal series 1 z K ′ (cid:16) z (cid:17) = Q ( z ) P ( z ) . Details of the algorithmic aspects of this derivation are due to Stieltjes. They aremasterfully exposed in Chapter 9 of [15]. It follows that F ′ ( − s ) F ( − s ) = K ′ ( s ) = s − Q ( s − ) P ( s − ) , and so we obtain the identity1 + m X k =1 f k ( µ , . . . , µ k )( − s ) k = s m P ( s − ) . We stress that this Pad´e approximation procedure identifies the denominator in themoment generating series without computing its zeros.Above, we touched on the old and eternally dominant theme of inversion of theLaplace transform. An effective method of inversion for rational functions withoutrelying on simple-fraction decompositions appears in [11]. For a general overview ofLaplace transform-inversion, we refer to the monograph [4].3. Proof of the main result With the results of the preceding section at hand, we now prove the main result ofthis paper. We first introduce some notation which will be used in the proof and in alater result. Definition 3.1. Given an integer m ≥ K of positive integers, we let∆ m ( K ) := [ k ∈ K ∆ m ( k ) , where ∆ m ( k ) := { j = ( j , . . . , j m ) ∈ Z m : j , . . . , j m ≥ , j + · · · + j m = k } is the set of integer-lattice points in the scaled standard m -simplex. For any k ≥ j = ( j , . . . , j m ) ∈ ∆ m ( k ) and any a = ( a , . . . , a m ) ∈ R m , we let (cid:18) k j (cid:19) = k ! Q ml =1 j l ! , a j := m Y l =1 a j l l , and j · a := m X l =1 j l a l . Proof of Theorem 1.4. Part (1). We note first that if the polynomial p is such that p (0) = 0 and Λ α is a Hirschman–Widder density for some α ∈ (0 , ∞ ) m , then p ◦ Λ α equals p (0) on ( −∞ , c Λ α is negative on(0 , ∞ ) if c < 0. Thus we need only to consider polynomials of degree at least two withzero constant term for the remainder of this proof.Suppose first that m ≥ 4. To construct the null set N ⊂ (0 , ∞ ) m , we begin asfollows. For any integer n ≥ 2, we let K n := { K ⊂ { , . . . , n } : n ∈ K } denote the set of subsets of { , . . . , n } containing n and, for any K ∈ K n , we define thenon-zero multivariate polynomial P K by setting P K = f − f , where f i ( a ) := ( − ( i − n V ( b a i ) n Y j ∈ ∆ m ( K ) \{ n e i } ( j − n e i ) · a ( a ∈ R m , i = 1 , , with b a = ( a , a , . . . , a m ) and b a = ( a , a , . . . , a m ), in accordance with Definition 2.5,and { e , e , . . . , e m } being the standard basis of R m .To see that P K = 0, we observe that the a -degree of f exceeds that of f , whichgives the claim. To see this, we note first that every factor in the product in f has alinear a -term, as n is the maximum element of K , sodeg a f = | ∆ m ( K ) | − . However, the Vandermonde determinant V ( b a ) contributes m − a -terms to f and therefore deg a f = n ( m − 2) + | ∆ m ( K ) | − |{ j ∈ ∆ m ( K ) : j = 0 }| = n ( m − 2) + | ∆ m ( K ) | − | ∆ m − ( K ) | . Using the fact that m ≥ n ≥ 2, it follows thatdeg a f − deg a f = | ∆ m − ( K ) | − n ( m − − ≥ | ∆ m − ( n ) | − n ( m − − (cid:18) n + m − m − (cid:19) − n ( m − − > ( n + m − · · · ( n + 2)( n + 1)( m − − ( n + 1)( m − ≥ ( n + 1) (cid:16) m ( m − · · · m − − ( m − (cid:17) = ( n + 1) (cid:16) m ( m − − ( m − (cid:17) = ( n + 1)( m − m − ≥ . Let Z K denote the zero locus of P K in R m , which is a null set because P K is non-zero. With S m denoting the group of permutations of { , . . . , m } , we let σ ∈ S m acton subsets of R m by permuting coordinates, so that σ ( A ) := { ( a σ (1) , . . . , a σ ( m ) ) : a = ( a , . . . , a m ) ∈ A } for any A ⊂ R m , IRSCHMAN–WIDDER DENSITIES 21 and note that this action is measure preserving. Finally, we let H q := { x ∈ R m : q · x = 0 } , (3.1) e N := (0 , ∞ ) m ∩ (cid:18) [ q ∈ Q m \{ } H q ∪ [ σ ∈ S m ∞ [ n =2 [ K ∈K n σ ( Z K ) (cid:19) (3.2)and N := { ( a − , . . . , a − m ) : a ∈ e N } . (3.3)As a countable union of null sets, the set e N is null. Furthermore, the set N is alsonull. To see this, we note that the self-inverse map f : (0 , ∞ ) m → (0 , ∞ ) m ; ( x , . . . , x m ) ( x − , . . . , x − m )is Lipschitz when restricted to [ l − , l ] m for any positive integer l , so preserves null setsthere, and N = ∞ [ l =1 f (cid:0) e N ∩ [ l − , l ] m (cid:1) . Let α ∈ (0 , ∞ ) m \N . Then the reciprocals of the entries of α are linearly independentover Q , since they are contained in no hyperplane of the form H q , so they are distinct.Thus, we may find a ∈ (0 , ∞ ) m \ e N such that a < · · · < a m , the sets { a − , . . . , a − m } and { α , . . . , α m } are equal and P K ( a ) = 0 for any n ≥ K ∈ K n .Now let c be as in Proposition 2.4, so that Λ a , c = Λ α . If c := a · · · a m V ( a )then d = ( d , . . . , d m ) := c − c = (cid:0) V ( b a ) , − V ( b a ) , . . . , ± V ( b a m ) (cid:1) . (3.4)Let Λ := Λ a , d = c − Λ α . If p is a polynomial of degree at least two such that p (0) = 0then p ◦ Λ α = q ◦ Λ, where the polynomial q is such that q ( x ) = p ( cx ), so q also hasdegree at least two and no constant term. Thus it suffices to show that p ◦ Λ is not aP´olya frequency function, where p ( x ) = P k ∈ K r k x k , with K ∈ K n for some n ≥ r k = 0 for all k ∈ K .For any non-negative integer k , the bilateral Laplace transform B{ Λ k } ( s ) = X j ∈ ∆ m ( k ) (cid:18) k j (cid:19) d j s + j · a = p k ( s ) q k ( s ) , where the notation is as in Definition 3.1 and q k ( s ) := Y j ∈ ∆ m ( k ) ( s + j · a ) . Thus B{ p ◦ Λ } ( s ) = X k ∈ K r k B{ Λ k } ( s ) = X k ∈ K r k p k ( s ) q k ( s ) = P ( s ) Q ( s ) , where Q ( s ) := Y k ∈ K q k ( s ) , b q k ( s ) := Y l ∈ K \{ k } q l ( s ) and P ( s ) := X k ∈ K r k p k ( s ) b q k ( s ) . For any k ∈ K , the polynomial q k has the set of roots {− k · a : k ∈ ∆ m ( k ) } . As theentries of a are linearly independent over Q , the roots of Q are simple. Furthermore,if k ∈ ∆ m ( k ) then a straightforward computation gives that P ( − k · a ) = r k (cid:18) k k (cid:19) d k Y j ∈ ∆ m ( K ) \{ k } ( j − k ) · a = 0 , (3.5)again using linear independence. Thus P does not vanish at any root of Q , and soTheorem 1.2 implies that p ◦ Λ is not a P´olya frequency function as long as P is notconstant. However, P ( − n e · a ) − P ( − n e · a )= r n (cid:18) V ( b a ) n Y j ∈ ∆ m ( K ) \{ n e } ( j − n e ) · a + ( − n +1 V ( b a ) n Y j ∈ ∆ m ( K ) \{ n e } ( j − n e ) · a (cid:19) = r n P K ( a ) = 0 , and this completes the proof whenever m ≥ m = 3. When p is a monomial, this was resolved inprevious work [3, Lemma 11.2] whenever the reciprocals of the entries of α are linearlyindependent over Q , so for Λ as above. It remains to verify that p ◦ Λ is not a P´olyafrequency function when p ( x ) = P k ∈ K r k x k , with r k = 0 for all k ∈ K and K ∈ K n containing at least two elements. In this case, the polynomial P K is non-zero, since wehave that deg a f − deg a f = | ∆ ( K ) | − n − > | ∆ ( n ) | − n − . Thus we may proceed as above, as long as we take only K containing at least twoelements in the definition of e N . Part (2). We consider first the case where α = · · · = α m = α . The correspondingHirschman–Widder density Λ α has bilateral Laplace transform B{ Λ α } ( s ) = (1 + αs ) − m , and inverting this transform gives thatΛ α ( x ) = x ≥ α − m ( m − x m − e − α − x . It follows immediately that, for any natural number n , the function Λ n α is a positivemultiple of Λ β , where β = ( αn − , αn − , . . . , αn − ) ∈ (0 , ∞ ) n ( m − has all its entriesequal, and so Λ n α is a P´olya frequency function.Now suppose that the polynomial p is not a positive multiple of a monomial. If p hasa constant term then p ◦ Λ α is not integrable, so we may assume that p ( x ) = P k ∈ K r k x k ,where K is a finite set of natural numbers with at least two elements and r k = 0 forall k ∈ K . Then B{ p ◦ Λ α } ( s ) = X k ∈ K r k c k (1 + αk − s ) − k ( m − − = P ( s ) Q ( s ) , IRSCHMAN–WIDDER DENSITIES 23 where c k > k ∈ K , Q ( s ) := Y k ∈ K (1 + αk − s ) k ( m − and P ( s ) := X k ∈ K r k c k Y j ∈ K \{ k } (1 + αj − s ) j ( m − . The polynomial P is non-constant, since each of the terms in the sum are polynomialswith distinct positive degrees. Furthermore, the roots of Q are of the form − kα − for k ∈ K , and P ( − kα − ) = r k c k Y j ∈ K \{ k } (1 − j − k ) j ( m − = 0 . Hence Q ( s ) /P ( s ) is not the restriction of an entire function and so p ◦ Λ α is not a P´olyafrequency function, by Theorem 1.2.It remains to consider the case where a j = a + ( j − δ for j = 1, . . . , m , where δ is positive and independent of j . We consider the case m = 2 first, so thatΛ n α ( x ) = x ≥ (cid:16) a a δ (cid:17) n n X j =0 (cid:18) nj (cid:19) ( − j e − ( na + jδ ) x (3.6)for any natural number n . If b := ( na , na + δ, . . . , na + nδ ) and d = ( d , . . . , d n ) issuch that Λ b , d is a Hirschman–Widder density, then Proposition 2.4 gives that d j = b j Y k = j b k b k − b j = 1 n ! δ n n Y k =0 ( na + kδ )( − j (cid:18) nj (cid:19) . Thus Λ n α is a positive multiple of the Hirschman–Widder density Λ b , d , so is itself aP´olya frequency function.For m ≥ 3, let β := ( b − , b − ), where b := a / ( m − 1) and b := b + δ = a m / ( m − m − β is a positive multiple of theHirschman–Widder density Λ α . Hence Λ n α is a positive multiple of the P´olya frequencyfunction Λ n ( m − β , and so is a P´olya frequency function itself.Finally, suppose that α /α is irrational and let p ( x ) = P k ∈ K r k x k , where K is afinite set of natural numbers with at least two elements and r k = 0 for all k ∈ K ; asnoted above, we need only consider p of this form. With the previous notation, wehave that Λ k α = c k Λ ( m − k β for some c > 0, and therefore B{ p ◦ Λ α } ( s ) = X k ∈ K r k c k ( m − k Y j =0 ( s + ka + jδ ) − = P ( s ) Q ( s ) , where c k = 0 for any k ∈ K , Q ( s ) := Y k ∈ K ( m − k Y j =0 ( s + ka + jδ )and P ( s ) := X k ∈ K r k c k Y v ∈ K \{ k } ( m − v Y u =0 ( s + va + uδ ) . The roots of Q are simple, since if ( j, k ) and ( j ′ , k ′ ) are distinct then ka + jδ = k ′ a + j ′ δ ⇐⇒ (cid:0) ( k − j ) − ( k ′ − j ′ ) (cid:1) a + ja = ( j ′ − j ) a = ⇒ α α = a a ∈ Q . It follows that if k ∈ K and j ∈ { , , . . . , ( m − k } then P ( − ka − jδ ) = r k c k Y v ∈ K \{ k } ( m − v Y u =0 (cid:0) ( v − k ) a + ( u − j ) δ (cid:1) = 0 . Furthermore, the polynomial P is the sum of polynomials with distinct positive degrees,so is non-constant. We conclude from Theorem 1.2 that p ◦ Λ α is not a P´olya frequencyfunction. (cid:3) We conclude with a discussion of the structure of the class of polynomials mappinga fixed Hirschman–Widder density into the class of P´olya frequency functions. Proposition 3.2. Suppose the entries of a = ( a , . . . , a m ) ∈ (0 , ∞ ) m are linearlyindependent over Q and strictly increasing, and let c be as in Proposition 2.4. Forthe polynomial p ( x ) = P k ∈ K r k x k , where K is a finite set of non-negative integersand r k = 0 for all k ∈ K , the following are equivalent. (1) p ◦ Λ a , c is a P´olya frequency function. (2) The function eval e P : ∆ m ( K ) → R ; k r | k | (cid:18) | k | k (cid:19) c k Y j ∈ ∆ m ( K ) \{ k } ( j − k ) · a is constant, where | k | := k + · · · + k m .Proof. Following the proof of Theorem 1.4(1), we have that B{ p ◦ Λ a , c } ( s ) = e P ( s ) /Q ( s ),where Q ( s ) = Y k ∈ ∆ m ( K ) ( s + k · a ) and e P ( s ) = X k ∈ ∆ m ( K ) r | k | (cid:18) | k | k (cid:19) c k Y j ∈ ∆ m ( K ) \{ k } ( s + j · a ) . As for P above, the polynomial e P does not vanish at any root of Q . Thus p ◦ Λ a , c isa P´olya frequency function if and only if e P is a constant. Since deg e P < | ∆ m ( K ) | , itsuffices to check that evaluating e P at the distinct points {− k · a : k ∈ ∆ m ( K ) } alwaysyields the same answer, but this is precisely (2) above. (cid:3) Remark 3.3. We may further characterize when a polynomial p ( x ) = P k ∈ K r k x k mapsthe Hirschman–Widder density Λ a , c into the class of all such densities. This happensif and only if Proposition 3.2(2) holds and the function p ◦ Λ a , c has unit integral. As( p ◦ Λ a , c )( x ) = X k ∈ ∆ m ( K ) r | k | (cid:18) | k | k (cid:19) c k e − ( k · a ) x , we obtain the additional condition X k ∈ ∆ m ( K ) r | k | (cid:18) | k | k (cid:19) c k k · a = 1 . IRSCHMAN–WIDDER DENSITIES 25 References [1] Michael Aissen, Albert Edrei, I. J. Schoenberg, and Anne Whitney. On the generating functionsof totally positive sequences. Proc. Natl. Acad. Sci. USA , 37(5):303–307, 1951.[2] S. V. Amari and R. B. Misra. Closed-form expressions for distribution of sum of exponentialrandom variables. IEEE Trans. Reliab. , 46(4):519–522, 1997.[3] Alexander Belton, Dominique Guillot, Apoorva Khare, and Mihai Putinar. Post-compositiontransforms of totally positive kernels. Preprint , arXiv:math.CA/2006.16213v3, 2020.[4] Alan M. Cohen. Numerical methods for Laplace transform inversion . Numerical Methods andAlgorithms 5. Springer, New York, 2007.[5] William Feller. 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On variation-diminishing integral operators of the convolution type. Proc. Natl.Acad. Sci. USA , 34(4):164–169, 1948.[20] I. J. Schoenberg. On P´olya frequency functions. II. Variation-diminishing integral operators ofthe convolution type. Acta Sci. Math. (Szeged) , 12(B):97–106, 1950.[21] I. J. Schoenberg. On P´olya frequency functions. I. The totally positive functions and their Laplacetransforms. J. d’Analyse Math. , 1:331–374, 1951.[22] A. Sharma and J. Tzimbalario. A class of cardinal trigonometric splines. SIAM J. Math. Anal. ,7(6):809–819, 1976. (A. Belton) Department of Mathematics and Statistics, Lancaster University, Lan-caster, UK Email address : [email protected] (D. Guillot) University of Delaware, Newark, DE, USA Email address : [email protected] (A. Khare) Department of Mathematics, Indian Institute of Science; and Analysis andProbability Research Group; Bangalore, India Email address : [email protected] (M. Putinar) University of California at Santa Barbara, CA, USA and Newcastle Uni-versity, Newcastle upon Tyne, UK Email address ::