NOTE ON APPELL POLYNOMIALS I. M. SHEFFER An interesting characterizatin f Appell plynmials by means f a Stieltjes integral has recently been given by Thrne. 1 We prpse t give a secnd such representatin, and t extend the result t the case f sets f plynmials f type zer, f which Appell sets frm a subclass. Appell sets may be defined by either f the fllwing equivalent cnditins: \P n {x)} 1 n = 0, 1,, is an Appell set (P n being f degree exactly n) if either (i) P n '(x)=pn-i{x),n = l,2, r (ii) there exists a frmal pwer series A(t) ^^a n t n (a^o) such that (again frmally) 00 A(t)e>* = X)P (*)/». The functin A (t) may be called the determining functin fr the set {P (x)}.the essence f Thme's result is the fllwing : THEOREM OF THORNE. A plynmial set {P n (x)} is an Appell set if and nly if there exists a functin a(x) f bunded variatin n (0, <*> ) with the fllwing prperties : (i) The mment integrals fx n = I 0 % n da(x) all exist. (ii) M^O. (iii) 00 P%\x)da(x) = ô nr, ô nr = lfr n=r, 5 nr = 0 fr n^r. And fr the set {P n (x)} the determining f unctin A(t) is given by The Stieltjes integral characterizatin that we nw give will be seen t be essentially different frm that in (iii) abve. THEOREM 1. A plynmial set {P n (x)} is an Appell set if and nly Presented t the Sciety, September 17, 1945 ; received by the editrs May 10, 1945. 1 C. J. Thrne, A prperty f Appell sets, Amer. Math. Mnthly vl. 52 (1945) pp. 191-193. 739
740 I. M. SHEFFER [Octber if there is a functin fi(x) f bunded variatin n (0, ) with the fllwing prperties : (i) The mment cnstants (1) b n ~ f x«df}(c6) all exist. (ii) i^o. (iii) Frn^O, 1,,» (x A- t) - ~-m^ n If (i) and (ii) hld, then P n (x) as given by (2) exists fr each w, and is a plynmial f degree exactly n. Mrever, differentiatin under the integral sign is permissible, s that P«(x) = P n _i(#) ; that is, {P n (x)} is an Appell set. Nw suppse {P n (x)} is an Appell set, and let A (t) be its determining functin : A (t) ~^T,?a n t n. Define the sequence {b n } by nl b n =* nla n. By a therem f Bas 2 there is a functin &(x) f bunded variatin n (0, ) whse mment cnstants are {b n } ; and since a^o, therefre ÔOJ^O. If we dente the right side f (2), which exists, by Q n (x) t then {Qn(#)} is an Appell set. Since (frmally)» 00 /» 00 «"<*+<W) - e"* I the determining functin fr {Q n (x)} is e ut dp(t), > / /» \ yn ^n ^Wj8(0 = El W)> - E *. w ) nl nl = E a «^n ^ A( u )- It fllws that {Q n (x)} s {P w (#)}, s that (2) hlds. COROLLARY. The determining f unctin f r {P n (x)} > 00 00 ^ <? W W0(/) - E ** n -»! 2 Widder, TTte Laplace transfrm, p. 139. This result f Bas is stated fr real sequences, but it extends immediately t cmplex sequences. is
1945] APPELL POLYNOMIALS 741 Let {J n (#)} be the Appell set (4) I n (x) - *»/*!; then (2) can be written (5) P n (x) = f I n (x + t)dp(t)- Nte that I 0 (0) = 1, 7 n (0)=0 (w>0); it is this prperty that makes the determining functin A (t) s simply expressible in terms f the mment cnstants {b n }. It is easily shwn that in relatin (5) the plynmials I n (x) can be replaced by any ther Appell set, by a suitable change in the functin j3(/). Hwever, the determining functin A{t) is nw nt s easily expressed thrugh the mment cnstants {b n } In fact, if {Q n (x)} is the Appell set denned by (6) Qn(x) «("Ux + Qdyit), then (7) P n (x) - f W+*)#(0, where the functin A (z) fr {P n (x)} with is given by e' 1 J e'*dy(u)dp(t) - a n z», J (9) a n = (l/nl) {<?»ft 0 + Cn.iCn-i&i + + C ntn c 0 b n }. Here {& n }, {c n } are given by» 00 / 00 x n dp{x), c n = I # n d7(tf),» = 0, 1,. We turn nw t sets f type zer. The plynmial set {P n (#)} ' ls f type zer 3 if either f the fllwing equivalent cnditins hlds : (i) Frmal series (11) A(t) = J % (<* * 0), H(t) «f) h n t n (h * 0) 8 1. M. Sheffer, Sme prperties f plynmial sets f type zer, Duke Math, J. vl. 5 (1939) pp. 590-622.
742 I. M. SHEFFER [Octber exist fr which (frmally) (12) A(t)e** «Y,Pn(x)t\ (ii) An peratr L \y{x) ] f frm (13) L[y(x)]^f^l n y^(x) (h ^ 0) l exists such that (14) L[P n (x)] = P n -i(*), n = 1, 2,.... If L(t) is the series (15) L(t) = W», l then T(/) and L(/) are (frmally) inverse functins: (16) L{E{t)) - H{L{t)) «/. All plynmial sets satisfying (14) will be said t be assciated with the peratr L. Sets assciated with a given L are distinguished ne frm anther by their determining f unctin A (t) appearing in (12). Assciated with a given L there is a unique set Bi 4 = {B n (x)} fr which J5(0) = 1, J3 W (0) =0 (n>0). This set we call the basic set fr L; its determining functin is A(t)^l. THEOREM 2. In rder that the plynmial set {P n (x)} be a set assciated with the peratr L it is necessary and sufficient that there exist a functin fi(x) f bunded variatin n (0, <*> ), with the fllwing prperties : (i) The mment cnstants {b n } fr p(x) all exist. (ii) &0 7*0. (iii) Frn = 0, 1,, (17) P n (x) = f B n (x + t)dfi(t), where \B n {x)} is the basic set fr L. has the ex COROLLARY. The determining f unctin A(t) fr {P n (x)} pressin (18) A(t) = f e«*mdfi(u) 9
19451 APPELL POLYNOMIALS 743 which is equivalent t ( 1 9 ) - - - - - -»» A(L(z)) = f e u dp(u) = J2 * n - t/ 0 w! If cnditins (i), (ii) hld, it is clear that P n (#)} as given by (17) is a set assciated with the peratr L. Nw let {P n {x)} be a set relative t an peratr L; t shw that (i), (ii), (iii) hld. Let A(t) be the determining functin fr {P n (x)}, s that we have relatin (12). Since A and L are given, the functin A(L(z)) is knwn; let its (frmal) pwer series be (20) A{L{z)) = 'Zatz\ 0 Nw define {b n \ s that & n = w!a*, and let j3(/) be the functin f bunded variatin n (0, <*> ), guaranteed by the Bas therem, whse mments are {b n }. With this /3( ), the right side f (17) defines a plynmial set {Q n (x)}, assciated with L, whse determining functin is seen t be the functin A(t) given by (18). As (18) is carried int (19) by the transfrmatin z=h(t) (and therefre t=l(z)) t it fllws that {P n }, {Qn} have the same determining functin A(t), and are therefre the same set. Hence (17) hlds. In Therem 2 the representatin (17) can be replaced by (21) P n (x) = f Q n (x + *)#(*). where {Qn(#)} is any plynmial set assciated with L; but in this case the determining functin A(t) fr {P (#)} has a mre cmplicated representatin than (18) r (19). In fact, if we write (22) Q n (x) = f B n (x + t)dy(t), 0 it is readily shwn that / I 00 f 00 I e<«+»>»w<*y(«)<*/3(w), 0 " O r, what is equivalent t this,» 00 /» 00 00 J where (25) a n = (l/«!) {c n b 0 + C n,ic n -ibx + + C n >&n}.
744 I. M. SHEFFER Here {b n }, {c n } are the mment cnstants fr j3(2), y(t) respectively. We clse with tw bservatins. REMARK 1. The Thrne therem carries ver t plynmial sets f type zer; we have nly t replace the cnditin (iii) f that therem by (26) M[P (*)]J«( where L (1) ==, L c2 \ are the iterates f L. Als, the determining functin A(t) fr {P n (#)} is given by (27) A(t) - [" f V*<'>Ja(*) r, by the equivalent expressin, (28) A(L(z)) - [ Vda(*)] ' - [ ^z»] \ REMARK 2. Thrughut this nte we have nt hesitated t use frmal pwer series. This des nt invalidate the results btained ; it has permitted the results t be gtten faster. As t hw we culd have reasned withut the use f these frmal series, we refer t the reference in ftnte 3, where there is t be fund n page 596 a nte f justificatin. THE PENNSYLVANIA STATE COLLEGE