On certain non-unique solutions of the Stieltjes moment problem

Transcription

1 Discee Mahemaics and Theoeical Compue Science DMTCS vol. 2:2, 2, On ceain non-unique soluions of he Sieljes momen poblem Kaol A. Penson Pawel Blasiak 2 Géad H. E. Duchamp 3 Andzej Hozela 2 Allan I. Solomon,4 LPTMC, CNRS UMR 76. Univesié Piee e Maie Cuie, Pais, Fance. 2 H. Niewodniczański Insiue of Nuclea Physics, Polish Academy of Sciences, Kaków, Poland. 3 LIPN, CNRS UMR 73. Insiu Galilée - Univesié Pais-Nod, Villeaneuse, Fance. 4 The Open Univesiy, Physics and Asonomy Depamen, Milon Keynes, Unied Kingdom eceived Augus 4, 29, evised Mach 8, 2, acceped Apil 5, 2. We consuc explici soluions of a numbe of Sieljes momen poblems based on momens of he fom ρ () (n) = (2n)! and ρ () 2 (n) = [(n)!]2, =, 2,..., n =,, 2,..., i.e. we find funcions W (),2 (x) > saisfying R x n W (),2 (x)dx = ρ(),2 (n). I is shown using cieia fo uniqueness and non-uniqueness (Caleman, Kein, Beg, Gu, Pakes, Soyanov) ha fo > boh ρ (),2 (n) give ise o non-unique soluions. Examples of such soluions ae consuced using he echnique of he invese Mellin ansfom supplemened by a Mellin convoluion. We ouline a geneal mehod of geneaing non-unique soluions fo momen poblems genealizing ρ (),2 (n), such as he poduc ρ () (n) ρ() 2 (n) and [(n)!]p, p = 3, 4,.... Keywods: Classical momen poblem, Sieljes momen poblem, Mellin ansfom Inoducion This pape concens soluions of he Sieljes momen poblem [, 2], i.e. posiive funcions W (x) which saisfy he infinie se of equaions x n W (x)dx = ρ(n), n =,, 2,.... () Coesponding auho: penson@lpl.jussieu.f c 2 Discee Mahemaics and Theoeical Compue Science (DMTCS), Nancy, Fance

2 296 K. A. Penson, P. Blasiak, G. H. E. Duchamp, A. Hozela, A. I. Solomon We peviously me his poblem when consideing he popeies of he so-called (genealized) coheen saes (CS) of Quanum Mechanics (i) defined, fo complex z D C and posiive numbes ρ(n), n =,, 2,..., as z = N /2 ( z 2 ) n= z n ρ(n) n (2) whee n s, n =,, 2,..., fom an ohonomal complee basis in he Hilbe space H, m n = δ mn, n= n n = (ii). These CS ae equied o be nomalizable, i.e. n= z 2n /ρ(n) = N ( z 2 ) has a non-zeo adius of convegence R, coninuous in he label z, i.e., z z implies z z, and o saisfy he popey of Resoluion of Uniy wih a measue µ( z 2 ) > fo z D C, whee D is a disc of adius R ceneed in z =, π D C d Re(z) d Im(z) µ( z 2 ) z z = = n n, (3) which is essenially equivalen o Eq. () fo W (x) = µ( z 2 )/N ( z 2 ) z 2 =x [3, 4]. The popey of Resoluion of Uniy plays an impoan ole in vaious applicaions of CS, in paicula in he consucion of he so-called Bagmann epesenaion in quanum mechanics wihin which quanum saes f = n= f n n, n= f n 2 = ae epesened as enie funcions [5, 6] f f B (z) = N /2 ( z 2 ) z f = n= n= f n z n ρ(n) (4) and he scala poduc of wo saes g f = n= g nf n is given in ems of so defined Bagmann funcions by g f = d Re(z) d Im(z) g π B(z)f B (z) µ( z 2 ) N ( z 2 ). (5) D C Taking g = m and f = n we see ha soluions o he momen poblem Eq. () povide us wih explici foms of he scala poduc in he Bagmann epesenaion geneaed by he CS of Eq. (2). Sandad CS, usually called Glaube o hamonic oscillao CS, fo which ρ(n) = n!, lead o µ( z 2 ) =, N ( z 2 ) = e z 2 and, consequenly, W (x) = e x. Genealized CS lead o ρ(n) s ohe han n! [7] and he soluions of Eq.(), if any, mus be sudied in each individual case sepaaely [8, 9,, ]. An efficien (i) Coheen saes wee oiginally poposed by Ewin Schödinge in he ealy days of Quanum Mechanics o descibe wave packes obeying he ime evoluion close o he classical moion (E. Schödinge, De Seige Übegang von de Miko- zu Makomechanik, Nauwissenschafen 4 (926) ). They wee einoduced moe han half a cenuy ago in seminal papes by R. J. Glaube and ohes: R. J. Glaube, The quanum heoy of opical coheence, Phys. Rev. 3 (963) , E. C. G. Sudashan, Equivalence of semiclassical and quanum mechanical descipion of saisical ligh beams, Phys. Rev. Le. (963) , and J. R. Klaude, Coninuous-epesenaion heoy. I. Posulaes of coninuous-epesenaion heoy, J. Mah. Phys. 4 (963) 55 58; Coninuous-epesenaion heoy. II. Genealized elaion beween quanum and classical dynamics, J. Mah. Phys. 4 (963) Such saes ae nowadays an impoan ool widely used in quanum opics and in geneal invesigaions of quanizaion. (ii) Hee we follow Diac s noaion univesally used in quanum physics: elemens of a Hilbe space ae denoed by kes, hei Hemiean conjugaes by bas, pojecion opeaos as and he scala poduc as.

3 On ceain non-unique soluions of he Sieljes momen poblem 297 way o ackle he poblem is o use he invese Mellin ansfom mehod which allows one o esablish many soluions of Eq. (), eihe by analyic mehods [2] o by exensive use of available ables [3, 4]. As a bypoduc of his mehod we have esablished ha, fo a lage numbe of combinaoial sequences such as Bell and Caalan numbes, ec., he coesponding sequences ρ(n) ae soluions of he momen poblem Eq. () [5, 6]. Likewise, sequences aising in heoy of odeing of diffeenial opeaos [7] solve appopiae momen poblems oo [8]. We wan also o emphasize ha any invesigaion of he momen poblem is deeply ooed in a classical poblem of saisics, namely ha of he unique o non-unique deeminaion of a pobabiliy disibuion fom is momens. We will link o i in Secion 4; hee we only emak ha his poblem was fully eaed moe han one hunded yeas ago by T. J. Sieljes [9], ecalled by M. G. Kein in he middle of he XXh cenuy [2] and is sill he subjec of exensive eseach which in ecen yeas has led o significan pogess in is undesanding [2, 22, 23, 24, 25, 26, 27, 28, 3, 29, 3, 32, 33, 34]. Saisical aspecs of he momen poblem have an analogue in quanum physics: accoding o is sandad inepeaion he pobabiliy ha he sae n appeas in he coheen sae z of Eq.(2) is P n (z) = n z 2 = z 2n N ( z 2 )ρ(n) (6) and, if ρ(n) saisfy Eq. () wih W (x) = µ( z 2 )/N ( z 2 ) z 2 =x. Then fo all n =,, 2,... we have D C d Re(z) d Im(z) µ( z 2 )P n (z) =, (7) which is equivalen o he Resoluion of Uniy of Eq.(3), asseing he compleeness of so defined CS. The physical inepeaion and consequences of uniqueness o non-uniqueness of he measue µ( z 2 ) saisfying Eq.(7) is howeve beyond he scope of he cuen pape and will be eaed elsewhee [35]. This admixue of quanum-mechanical, analyical, combinaoial and saisical feaues deseves a deepe sudy which we inend o pusue. Ou pape is paly exposioy in chaace, and has he following sucue: fis we esablish a link beween he Mellin ansfom and he momen poblem; nex, in Secion 3, we povide pincipal soluions o wo momen poblems, emed oy models. In Secion 4 we discuss cieia fo he uniqueness of soluions of he momen poblem. Secion 5 is devoed o he explici consucion of non-unique soluions of he oy-models. In Secion 5, some genealizaions of model sequences, ogehe wih hei soluions, ae eviewed. Secion 6 is devoed o a discussion and conclusions. We dedicae his pape o Philippe Flajole on he occasion of his 6h bihday. His pioneeing applicaions of Mellin ansfom asympoics o he analysis of combinaoial sucues [36, 37, 38] have been a souce of inspiaion fo us.

4 298 K. A. Penson, P. Blasiak, G. H. E. Duchamp, A. Hozela, A. I. Solomon 2 The Mellin ansfom vesus momen poblem The Mellin ansfom of a funcion f(x) of he eal vaiable x is defined fo complex s by he following elaion M[f(x); s] = x s f(x)dx f (s) (8) ( in his definiion f (s) is no he complex conjugae of f(s)! ), and is invese M is defined by M [f (s); x] = 2πi c+i c i f (s)x s ds. (9) See Ref. [39] fo a discussion of he dependence of f(x) on he eal consan c. Among he many elaions saisfied by he Mellin ansfom we shall mainly use he following M[x b f(ax h ); s] = ( ) s+b a h f s + b, a, h >. () h h If M[f(x); s] = f (s) and M[g(x); s] = g (s), hen M [f (s)g (s); x] = f ( x ) g() d = g ( x ) f() d () which is called he Mellin convoluion popey. Noe ha if in Eq. () boh f(x) and g(x) ae posiive fo x > hen f( x d )g() is also posiive fo x >. This means ha he Mellin convoluion peseves posiiviy, an essenial popey when consideing he momen poblem. Eq.(), if ewien fo n = s as W (x) = M [ρ(s ); x], (2) is, if W (x) >, a soluion of a Sieljes momen poblem. Thus accoding o Eq. (2) one can solve he Sieljes momen poblem by pefoming he invese Mellin ansfom on he momen sequence and checking if he esuling funcion is posiive. All he soluions in he sequel have been obained using Eqs. (2), () and (). Noe ha W (x) obained via Eq. (2) may no be he only soluion of Eq.(). We call W (x) > obained via Eq. (2) fom Eq. () he pincipal soluion (iii). (iii) Ses of posiive funcions consising of cene of he class modulaed by an oscillaoy funcion ohogonal o all polynomials, i.e., ses of posiive funcions shaing he same (Sieljes) momens, called a Sieljes class, wee fis discussed in [3] and hen, in vaious aspecs, consuced and analysed in [28, 32, 33].

5 On ceain non-unique soluions of he Sieljes momen poblem Pincipal soluions of he momen poblems Le us conside wo sequences of ineges given by ρ () (n) = (2n)!, n =,, 2,..., =, 2,.... (3) ρ () 2 (n) = [(n)!]2, n =,, 2,..., =, 2,.... (4) In he following we shall obain he soluions of he Sieljes momen poblem fo he momen sequences given by Eqs. (3) and (4), i.e. he funcions W (),2 (x) > saisfying x n W (),2 (x)dx = ρ(),2 (n). (5) Fom now on we shall efe o he model poblems ρ (),2 (n) as oy models TM and TM2, especively. a) TM: we begin by obaining W () (x). Noe ha (2n)! = Γ(2n + ) = Γ(2(s 2 )). We now apply Eq.() wih a =, b = 2 and h = 2. We subsequenly use M [Γ(s); x] = e x which gives [ ] x n W () (x)dx = x n e x 2 dx = (2n)!, n =,, 2,.... (6) x In he same spii we obseve ha (2n)! = Γ ( 2 ( )) ( ( )) n + 2 = Γ 2 s 2 2. Upon using Eq. () bu now wih a =, b = 2 2 and h = 2 one obains [ ] x n W () (x)dx = x n e x 2 dx = (2n)!, n =,, 2,.... (7) 2x 2 2 This means ha W () (x) > is he pincipal soluion of he momen poblem Eq. (7). b) TM2: we begin by deiving W () 2 (x) = M [Γ 2 (s); x] and employ he Mellin convoluion Eq. (). By using he Sommefeld epesenaion of he modified Bessel funcion of second kind K (x) [4] one obains x n W () 2 (x)dx = x n [ 2K (2x 2 ) ] dx = (n!) 2, n =,, 2,.... (8) Subsequenly noe ha [(n)!] 2 = [Γ ( ( )) n + ] 2 = [Γ ( ( )) s ] 2 and again apply Eq.() wih a =, b = and h =. The esul is [ ] x n W () 2 (x)dx = x n 2 K (2x 2 ) dx = [(n)!] 2, n =,, 2,.... (9) x Since K () > fo >, W () 2 (x) > is he pincipal soluion of he momen poblem Eq. (9). We emphasize ha alhough he invese Mellin ansfom echnique deseves o be bee known o boade physics communiy, i is sandad in pobabiliy [23, 4, 42].

6 3 K. A. Penson, P. Blasiak, G. H. E. Duchamp, A. Hozela, A. I. Solomon 4 Cieia of uniqueness and non-uniqueness of he Sieljes momen poblem As we emaked in he Inoducion i was ealized fom he vey beginning of he hisoy of he momen poblem ha is soluions may no be unique; i.e. fo a given momen sequence hee may exis moe han one soluion. Sieljes himself gave an example of a non-unique soluion of a poblem leading o wha uned ou o be he lognomal disibuion [9]. This example, being abou he only one available, was quoed epeaedly in he lieaue. As ecenly as weny yeas ago new non-unique soluions of ohe ypes of poblems have been consuced. Of special inees wee Sieljes momen poblems aising in pobabiliy heoy, elaed o invesigaion of pobabiliy disibuions no deemined by hei momens. Consequenly, he subjec was fuhe developed and sysemaized, lagely due o he compehensive wok of Beg [2, 23], Beg and Pedesen [22], Gu [24], Lin [25], Pakes wih cowokes [26, 27, 28], Soyanov [29, 3, 3], Soyanov and Tolmaz [32, 33] and ohes. Fo ecen exension of Ref. [2] see Osovska and Soyanov [34]. Fom he pacical poin of view one needs cieia o decide whehe he pusui of non-unique soluions is easonable. Such cieia ae eihe based on he ρ(n) s alone o on he soluion W (x) alone, o on boh ρ(n) and W (x), see below. Fom now on we assume ha all he ρ(n) s ae finie and ha W (x) is coninuous. We now give, in a somewha condensed fom, a lis of such cieia. C Caleman uniqueness cieion (T. Caleman, 922, []) This is based on he popeies of he ρ(n) s alone and is: If S = n= [ρ(n)] 2n =, hen he soluion is unique. This cieion does no imply ha if S < hen he soluion is non-unique. In fac i is possible o consuc models fo which S < and soluions ae sill unique [29, 43]. C2 Kein s non-uniqueness cieion (M. G. Kein, aound 95, [2]) This is based eniely on he soluion W (x) and does no involve he momens. If ln [W (x 2 )] +x 2 dx <, i.e., he so-called Kein inegal exiss, hen he soluion is non-unique. C3 Convese Caleman cieion fo non-uniqueness (A. Pakes, 2, [26], A. Gu, 22, [24]) This is based on ρ(n) s and W (x). If hee exiss x such ha fo x > x, < W (x) < and ψ(y) = ln [W (e y )] is convex in (y, ), whee y = ln (x ), and if, in addiion, S = n= [ρ(n)] 2n <, hen he soluion W (x) is non-unique. See [2, 22, 24, 26, 27, 29] fo vaious efinemens of hese cieia. 5 Consucion of non-unique soluions fo TM and TM2 We fis apply he cieion C o sequences ρ (),2 is conclusive) and conclude ha fo = boh soluions W () (x) = e ae unique. Fo > we find ha n= [ρ(),2 (n)] 2n (n) (he logaihmic es of divegence of he seies S x 2 x and W () 2 (x) = 2K (2x 2 ) is convegen. Applicaion of cieion C2 gives

7 On ceain non-unique soluions of he Sieljes momen poblem 3 convegence of he Kein inegal, showing ha he soluions of TM and TM2 ae non-unique. These findings ae confimed by use of cieion C3: convexiy of ψ (),2 (x) = ln [W (),2 (ex )] is poved as i is equivalen o e x/ > fo x >, (TM) and K (2e x/ ) K (2e x/ ) > fo x >, (TM2). The ques fo non-unique soluions fo > is hen well founded. We emak ha he phenomenon of swiching fom unique o non-unique soluions can be also explained by mehods exposed by Pakes e al. in Ref.[27], which would also pemi he sudy of non-inege case. We know of no geneal mehod o consuc such soluions. Howeve we have poposed a pocedue based on he applicaion of invese Mellin ansfom which geneaes he equied soluions [9, 44] which we now biefly expose. The fis sep is o consuc, wihin he famewok of a given se of ρ(n) s, a family of funcions ω k (x), paameized by a consan k (o be defined below), such ha all hei momens vanish, i.e. x n ω k (x)dx = x s ω k (x)dx =, n =,, 2,..., s =, 2,.... (2) The Mellin ansfom of ω k (x), ωk (s), vanishes fo s =, 2,.... Such funcions ae ohogonal o all polynomials and play an impoan ole in he sudy of inegal ansfoms [45]. Fo ou puposes we choose a paicula mehod of poducing he funcions ωk (x): o equivalenly x n ω k (x)dx = ρ (),2 (n) h k(n), (2) x s ω k (x)dx = ρ (),2 (s ) h k(s ), (22) whee h k (s) is any holomophic funcion vanishing fo s =, 2,.... Among an infiniy of possible choices he simples one is h k (s)=sin (πk(s + )) and i defines a discee paamee k =±, ±2, ±3,.... The funcion ω k (x) acquies new paamees now and is fomally obained by calculaing he invese Mellin ansfom ω (),2,k (x) = 2πi i i ρ (),2 (s ) sin (πks)x s ds, k = ±, ±2, ±3,.... (23) I uns ou ha fo boh ρ (),2 (s) he inegaion in Eq.(23) can be pefomed: a) TM: fo ρ () (n) = (2n)!, ω(),k [ ω (),k (x) = e x 2 sin 2x 2 2 [ = W () (x) sin (x) is a special case of ealie evaluaion [9] and eads: kπ ( 2 2 kπ ( 2 2 ) + x 2 an ( kπ 2 ) ], ( > k ) ) + x 2 an ( kπ 2 ) ]. (24)

8 32 K. A. Penson, P. Blasiak, G. H. E. Duchamp, A. Hozela, A. I. Solomon In Eq.(24) we noice a pleasan facoizaion of W () (x). b) TM2: fo ρ () 2 (n) = [(n)!]2 he coesponding funcion ω () 2,k (x) has o be, in he fis place, epesened as ω () 2,k (x) = M [Γ(s s + ) Γ(s s + ) sin (πks); x] (25) }{{}}{{} I II which can be conceived as anohe case of Mellin convoluion. A lile hough gives he wo panes o be convolued as I W (/2) (x) = e x, (26) x (see Eq.(24)) and [ II ω (/2),k (x) = e x sin kπ ( x (compae Eq.(24)). Thus, he inegal fom of ω () 2,k (x) is 2,k (x) = ω () W (/2) ) + x ( x ) ω (/2),k () d an ( kπ ) ], (27) whose evaluaion equies a numbe of changes of vaiables as well as he use of fomula , p. 453 of vol. Ref.[3], bu is essenially elemenay. The final esul is ω () 2,k (x) = 2 Re [e iπ( ( x 2 k ) ( ( K 2x 2 + i an πk )) /2 )] (29) 2 x V () k (x) whee we noe a nea facoizaion of W () 2 (x). Amed wih explici foms fo ω (),k (x) and ω() 2,k (x) we ae in posiion now o wie down families of non-unique soluions. Thei sucue has he fom: pincipal soluion + cons ω k (x). Moe pecisely: TM: [ ( ( ) ( ))] W () (ɛ, k, x) = W () 2 (x) + ɛ sin kπ + x kπ 2 an (3) 2 2 fo eal ɛ, ɛ <. TM2: fo [ > 2 k. As + γ V () k (x) K (2x /2 ) W () 2 (γ, k, x) = W () 2 (x) [ + γ V () k (x) K (2x ] is an oscillaing funcion of bounded vaiaion, a consan γ = γ(k, ) can be W () always found o assue he oveall posiiviy of 2 (γ, k, x). The above echnique fo obaining nonunique soluions can be eadily exended o momen sequences moe geneal han ρ (),2 (n). We shall simply menion wo such exensions wihou eneing ino deails. 2 ) ] (28) (3)

9 On ceain non-unique soluions of he Sieljes momen poblem 33 Fo ρ () 3 (n) = [(n)!]3 we begin wih he sequence (n!) 3 fo which he soluion is x n MeijeG ([ [ ], [ ] ], [ [,, ], [ ] ], x) dx = (n!) 3, (32) whee we use a convenien and self-explanaoy noaion fo Meije s G-funcion boowed fom ha of compue algeba sysems. Obseve ha ρ () 3 (n) coesponds o iple convoluion of pobabiliy disibuion funcion chaaceizing (n)!, hence he use of Meije s G-funcion. Fo applicaions of Meije s G funcion in pobabiliy heoy see Refs.[4, 42]. The exension, via Eq. (), leads o he pincipal soluion [ x n ( ) ] MeijeG [ [ ], [ ] ], [[,, ], [ ]], x dx = [(n)!] 3. (33) x The inegand in Eq. (33) is a posiive funcion fo x > which canno be epesened by any ohe known special funcion. I possesses an infinie seies epesenaion in ems of polygamma funcions, which we will no quoe hee. The Caleman sum S is convegen bu he cieion C2 is no conclusive. Only he cieion C3 pemis o asceain he non-uniqueness. The coesponding funcion ω () 3,k (x) is defined as 3,k (x) = M [Γ 2 (s s + ) Γ(s s + ) sin (πks); x] (34) }{{}}{{} I II ω () which can be calculaed as Mellin convoluion of I W () 2 (x) and II ω (/2),k (x), see Eqs. (24) and (29), especively. As a final example conside he sequence ρ () 4 (n) = ρ() (n)ρ() 2 (n). The coesponding Mellin convo- (x), see Eqs. (3) and (3), yields diecly he pincipal soluion fo luion of wo pincipal soluions W (),2 ρ () 4 (n): [ x n 4 πx 2 ( ) ] MeijeG [ [ ], [ ] ], [[ 2,,, ], [ ]], 4 x dx = (2n)![(n)!] 2, which is non-unique by C3 only, as C2 emains inconclusive. 6 Discussion and Conclusion n =,, 2..., =, 2,..., (35) We have demonsaed a mehodology fo obaining unique and non-unique soluions of he Sieljes momen poblem using he Mellin convoluion mehod. Alhough he iniial momen sequences wee simple and classical, one is apidly foced o leave he ealm of sandad special funcions, as he esuling soluions ae special cases of Meije G-funcions, a well-known fac o specialiss in pobabiliy and saisics. In mos cases hey esis he check fo non-uniqueness via boh he Caleman cieion C and he Kein cieion C2, and Pakes-Gu cieion C3 appeas o be he only ool o decide his quesion. We have geneaed paameized families of non-unique soluions exemplified hee by Eqs. (24) and (29). Such funcions, afe Soyanov [3], ae now called Sieljes classes. Thei geneal popeies and mehods

10 34 K. A. Penson, P. Blasiak, G. H. E. Duchamp, A. Hozela, A. I. Solomon of explici consucion fo such pobabiliy disibuions like lognomal, invese Gaussian and logisic, all leading o indeeminae Sieljes momen poblem, wee invesigaed by Soyanov and Tolmaz and Pakes [32, 33, 28]. In he conex of wok of he afoemenioned auhos ou example Eq. (24) belongs o he class aising fom Eq.(4) in [33] while he example Eq. (29) is moe geneal. This suppos ou belief ha he Mellin ansfom/convoluion echnique, which we have pesened and advocaed houghou his pape, povides a oolki compleing ohe mehods of solving he indeeminae momen poblems. Acknowledgemens We hank A. Gu, B. Chemin and H. L. Pedesen fo discussions and anonymous efeees fo commens. We also wish o acknowledge suppo fom Agence Naionale de la Recheche (Pais, Fance) unde Pogam No. ANR-8-BLAN and fom PAN/CNRS Pojec PICS No.4339 (28-2). Two of us (P.B. and A.H.) wish o acknowledge suppo fom Polish Minisy of Science and Highe Educaion unde Gans Nos. N and N /2832. Refeences [] N. I. Akhieze, The Classical Momen Poblem (Olive and Boyd, Edinbugh and London, 963). [2] B. Simon, The classical momen poblem as a self-adjoin finie diffeence opeao, Adv.Mah. 37 (998) [3] J. R. Klaude and E. C. G. Sudashan, Fundamenals of Quanum Opics (Benjamin, New Yok, 968). [4] J. R. Klaude and B.-S. Skagesam, Coheen Saes. Applicaion in Physics and Mahemaical Physics (Wold Scienific, Singapoe, 985). [5] V. Bagmann, On a Hilbe space of analyic funcions and an associaed inegal ansfom, Comm. Pue and Appl. Mah. 5 (96) [6] A. Voudas, Analyic epesenaions in quanum mechanics, J. Phys. A: Mah. Gen. 39 (26) R65 R4. [7] A. O. Bau and L. Giadello, New coheen saes associaed wih non-compac goups, Comm. Mah. Phys. 2 (97) [8] K. A. Penson and A. I. Solomon, New genealized coheen saes, J. Mah.Phys. 4 (999) [9] J.-M. Sixdenies, K. A. Penson and A. I. Solomon, Miag-Leffle coheen saes, J. Phys. A: Mah. Gen. 32 (999) [] J. R. Klaude, K. A. Penson and J.-M. Sixdenies, Consucing coheen saes hough soluions of Sieljes and Hausdoff momen poblems, Phys. Rev. A 64 (2) 387. [] K. A. Penson and A. I. Solomon, Coheen saes fom combinaoial sequences, in Poceedings 2nd In. Symposium Quanum Theoy and Symmeies, Kaków, Poland 8-22 July 2, pp (Wold Scienific, Singapoe, 22), axiv:quan-ph/5.

11 On ceain non-unique soluions of he Sieljes momen poblem 35 [2] O. I. Maichev, Handbook of Inegal Tansfoms of Highe Tanscenal Funcions, Theoy and Algoihmic Tables (Ellis Howood, New Yok, 983). [3] A. P. Pudnikov, Yu. A. Bychkov and O. I. Maichev, Inegals and Seies, vol., Elemenay funcions, vol.2, Special funcions, vol.3, Moe special funcions (Godon and Beach, New Yok, 998). [4] F. Obeheinge, Tables of Mellin Tansfoms (Spinge Velag, Belin,974). [5] K. A. Penson, P. Blasiak, G. Duchamp, A. Hozela and A.I. Solomon, Hieachical Dobiński elaions via subsiuions and he momen poblem, J. Phys. A: Mah. Gen. 35 (24) [6] P. Blasiak, A. Hozela, K. A. Penson and A. I. Solomon, Dobiński-ype elaions: Some popeies and applicaions, J. Phys. A: Mah. Gen. 37 (26) [7] P. Blasiak, A. Hozela, K. A. Penson, A. I. Solomon and G. H. E. Duchamp, Combinaoics and Boson nomal odeing: A genle inoducion, Am. J. Phys. 75 (27) [8] K. A. Penson, P. Blasiak, A. Hozela, G.H.E. Duchamp and A.I. Solomon, Laguee-ype deivaives: Dobiński elaions and combinaoial ideniies, J. Mah. Phys. 5 (29) [9] T. J. Sieljes, Recheches su les facions coninues, Anns. Fac. Sci. Univ. Toulouse 8 (894) J-J22, 9 (895) A5-A47. [2] M. G. Kein fomulaed logaihmic inegal cieia fo he Hambuge momen poblem in 4 and 5 of he XXh cenuy. Exensions of Kein s cieia o he Sieljes momen poblem wee found much lae. [2] C. Beg, Indeeminae momen poblems and he heoy of enie funcions, J. Comp. Appl. Mah., 65 (995) [22] C. Beg and H. L. Pedesen, Logaihmic ode and ype of indeeminae momen poblems, in Diffeence Equaions, Special Funcions and Ohogonal Polynomials, Poceedings of he Inenaional Confeence, Munich, Gemany 25-3 July 25, pp. 5 8, (Wold Scienific, Singapoe, 27). [23] C. Beg, On powes of Sieljes momen sequences, I, J. Theo. Pob., 8 (25) [24] A. Gu, On he momen poblem, Benoulli 8 (22) 47 42, and efeences heein. [25] G. D. Lin, On he momen poblem, Sa. Pob. Le. 35 (997) [26] A. G. Pakes, Remaks on convese Caleman and Kein cieia fo he classical momen poblem, J. Ausal. Mah. Soc. 7 (2) 8 4, and efeences heein. [27] A. G. Pakes, W-L. Hung and J-W. Wu, Cieia fo he unique deeminaion of pobabiliy disibuions by momens, Aus. N. Z. J. Sa. 43 (2), -. [28] A. G. Pakes, Sucue of Sieljes classes of momen-equivalen pobabiliy laws, J. Mah. Anal. Appl. 326 (27)

12 36 K. A. Penson, P. Blasiak, G. H. E. Duchamp, A. Hozela, A. I. Solomon [29] J. Soyanov, Couneexamples in Pobabiliy, 2nd ed., Secion (Wiley, New Yok, 997). [3] J. Soyanov, Kein condiion in pobabilisic momen poblems, Benoulli 6 (2) [3] J. Soyanov, Sieljes classes fo momen-indeeminae pobabiliy disibuions, J. Appl. Pob. 4 (24) [32] J. Soyanov and L. Tolmaz, New Sieljes classes involving genealized gamma disibuions, Sa. Pob. Le. 69 (24) [33] J. Soyanov and L. Tolmaz, Mehod fo consucing Sieljes classes fo M-indeeminae pobabiliy disibuions, Appl. Mah. Comp. 65 (25) [34] S. Osovska and J. Soyanov, A new poof ha he poduc of hee o moe exponenial vaiables is momen-indeeminae, Sa. Pob. Le. (2), in pin. [35] P. Blasiak, A. Hozela, K. A. Penson, A.I. Solomon and G.H.E. Duchamp, Genealized coheen saes: esoluion of uniy and indeeminacy of he momen poblem, in pepaaion. [36] P. Flajole, X. Goudon and P. Dumas, Mellin ansfoms and asympoics: Hamonic sums, Theo. Comp. Sc. 44 (995) [37] P. Flajole and R. Sedgewick, Mellin ansfoms and asympoics: Finie diffeences and Rice s inegal, Theo. Comp. Sc. 44 (995) 24. [38] P. Flajole and R. Sedgewick, Analyic Combinaoics (Cambidge Univesiy Pess, Cambidge UK, 29), Appendices B and C. [39] I. N. Sneddon, The Use of Inegal Tansfoms (Taa McGaw, New Delhi, 972). [4] N. N. Lebedev, Special Funcions and Thei Applicaions (Dove, New Yok, 972). [4] M. D. Spinge, The Algeba of Random Vaiables (Wiley, New Yok, 979). [42] A. M. Mahai, A Handbook of Genealized Special Funcions fo Saisical and Physical Sciences (Oxfod Univesiy Pess, Oxfod, 993). [43] see Ref. [2], emak on p.89, Coollay 4.2 and Theoem 6.2. [44] J.-M. Sixdenies, Consucions de nouveaux éas cohéens à l aide de soluions des poblèmes des momens, Thèse de Docoa de l Univesié Pais 6 (2), unpublished. [45] S. G. Samko, A. A. Kilbas and O. I. Maichev, Facional Inegals and Deivaives, Theoy and Applicaions (Godon and Beach, New Yok, 993).