Journal of Research of the National Burea u of Standards-B. Mathematics and Mathematical Physics Vol. 63B, No. 1, July- September 1959
Relations Between Summation Methods and Integral Transformations 1
Werner Greub 2
(M ay 12. 1959) R elations between the Lototsky method of summation and t ho se of Borel, Euler, a nd Knopp are obtained by associating an integral transformation wit h t he series transformation.
1. Introduction 2. General Relations Between Series and Integral Transforms In a r ecent paper , R . P . Agnew [1] 3 es tablishes re lations between the Lototsky method of summation for divergen t series and the classical summations of Let an, (n,v= 1,2, ... ) be an infinite matrLx whose Borel, Euler, and Knopp. clemen ts arc complex and subject to th e co ndition It is the purpose of t hi s paper to show that these relations can be ob tain ed in a very natural way if the lim ·~/ l anvl= O (n= 1, 2, . .. ) . (1) series tra nsformation is associated with an integral .->'" transformation . The following scheme describ cs the connections between the series and the integral We consider a llOt necessa rily convergen t infinite transforma tion : senes original series ~ transfo rmed seri es '" ~ U. (2) . =1 IL auren t expansion 1T aylor expansion wi th the proper ty analy tic fun ction ~ in tegral transform (3) of t be function In section 2 it is shown that und er certain con This cOllditioH assUl' C' s tha L Lite rad ill S 0 f convergence di tions a regular seri es tran sformati on can be of the power series ass igned to every integral transforma tion , and con versely. As an example, the integral transformation ± U. corresponding to the Loto tsky seri es transformation . =1 Z, (L-transformation) is constr ucted . This integral is positive. transfo rmation is used in section 3 for the di scussion Under cO:lldi tiolls (J ) and (3) all the series of the relation betwcen the Lo totsky and th e Borel integral methods. The ou tcome of this discussion is 00 that L-summability contains Borel integra l-summa Un= ~ anvuv bihty (BI-summability) but docs not contain a .=1 modified Borel sllmmability (BI*-summability as denoted in [1]). It is shown that the L-method can arc convergen t, as can readily be seen with th e help be ext ended to a summation method which is equiva of the root test. Ther efore, we can say that lInd er lent to the BI*-method. those conditions the matrix (an.) defines the series In section 4 it is proved that a power series is tl'ansfol'ma tion L-summable in all interior points of the Borel polygon '" U n---'7U n= ~ anvu v • (4) of t he corresponding analytic function but not v= 1 L-sllmmable in all the exteri or points. This proof is based on the relationship between the L-method We will show now th a t one can associa te witlt thi s 1 a nd the BI-methocl. series transforma tion a certain integral transforma In secti on 5 the Lototsky method is compared tion b etween the analy tic functions with t he method of Euler-lCn opp. In the last section the in verse of the L-integral '" u transform is constructed and Llsed to obtain the f (z) = ~ ,; (5) inverse m atrix of the Lototsky series transformation. v= 1 Z and 1 'r his wo rk was carried ou t in part under a Nation al B ureau of Standards 00 contract with T he America n University. n 2 P resent address: U n ivers it. y of Ztir"ic h, Zurich, Swi t zerland. ¢ ( a)= ~ U na . 3 Figures in brackets ind icate the literature references a t the end of this paper. n= l 1 To obtain the kernel of this transformation we From (7 ) we obtain between the coefficients U v and define the functions P ,, (z) by Un the rela tions
co 1 1 p ,Jz) = L::; an V+ I Z ' (n = l , 2, . .. ). rf: t/>(a) ( )2rf:rf: K(z,a) n ~ O U n= 27r'i J an+1 da= 27ri J'j an+ 1Z' j(z)dzda
Because of (1 ) these series are convergent for every = ( _1 "K(z ,a) v d Z· . )2pp L..; n+ l U• zea. (8) z and thus P ,, (z) are entire functions of z. 27r' ~ v a Z It follows from (3) that the functionj(z) is regular outside of a certain circle, Izl= p. Now we integrate Hence the integral transformation (7) defines a series the product, P n(z)j(z) along a circle Izl=R (R > p), transformation whose matrix is given by and obtain 1 rhrh K(z,a) l d t/>P,,(z)J (z) dz=rh L::; a "V+ lu vz '~d z nv ( )2 J V,jl a = 27ri 'j 'j an+ 1z' e Z a. = 27r'i ~ a n ~11~ = 27r'iUn ' (6) I' ~ l It can be shown that under certain hypotheses about ' the kernel the series transformation (8) is regular in UndeI' the h)'pothcsis that the function the following sense: If the series :L:uv is convergent, v the series L::; Un is Abelian summable (A-summable). n K(z, a)= ~ P n(z)an n ~ l This means that the limit is regular III th e domain Izl< oo, la l< l , we derive (a real) from (6)
exists and the relation
Hence, ddilling the function t/>(a) for la l t/> (a) = ~ unan, n ~ J holds. The hypotheses about the kernel are: (i ) K(z,a) is a regular function of z and a in the we obtain thc integral transformation 4 domain Izl < 00, Ia l< 1 and therefore can be expanded in the form '" K(z, a)J(z)dz, (7) K (z ,a) = :L: an+ l(a)Zn. (9) ( a)= 2~i ~ n~O which maps every function j(z) which is regular (ii) For every fixed n we have outside of a certain circle into a function ( a) which is regular within the unit circle. lim an (a)=l (a real) . Conversely, let (7) be an integral transformation <>-71- 0 whose kernel K(z, a) is regular in the domain Izl< 00 , !a l 2 U sin g a wcll -known I'cgul al'i Lv Lh eor em we concl udc scn es ~ U n is convcrgcnL rcspectivcl.\T A-summable from conditions (ii) and (iii) Lh at this transformation n is regul ar ; th is m cans, th c limit lim 4>(0' ) exists pro- the origin al series ~, ?j" will bc call cd L- ummable a - H-O v idecl Lhat th c series ~u , is conver gent and the r cspcctively AL-summable, and wc wri tc r elalion 00 lim 4> (0') = ~ u,. (11 ) a - H - O v=l and h olds. lim (~ U nan) = AL (~ u ,, ). Thc statement which has just becn provcd can bc a ~ 1-0 n v formulated as the following: TlmoR EM 1. L et K (z,a) be a function with the 00 For the integral kernel, K (z,a)= L. P ,,(z) a", of the properties (i), (ii), and (iii), and let the matrix anv n=l be defined by L-transformation, the functions 1-',,(z) JIaVC thc rol lowing form: 1 )2rh rh K (z ,a) a",= ( 27ri J J an+ 1zv dzela. P 1(z ) =Z Then the series transformation induced by this matrix 1 (_ 1) 11 (l -Z ) carries every convergent sel'ies ~ 1L , into a11 A -summable P,,(z)=rJ P n- I (z)= z - 1 n (n= 2, :) , ... ). v series ~ Un with the same sum. (15) n It is worth notin g lha t thc sc ri cs ~ Un n ee d not This lcads to 11 be convergent itself. However, if the mat ri x an, fulfills additional ill equalities of the form K (z, a) = _ 1_ ± (_ I )" (l -Z ) a ,,=~- a )I -Z - I . z-1 n= 1 n z- l "';;-"L..J 1a nv I= ~ U,,= lim ~ (U na") . n a -tl - O n AL (~ u v) = lim cf> (a). v a~ ! - O Thus, undcr th e additional h ypothesis (12), wc gct In ordcr to show that a convel'gcnt seri cs is also AL-summable to the sam e value we check that th e kernel K (z, a) satisfi es the h ypothesis of theol'cm 1. Condition (i) is immediatel.\T cl ear and (ii ) follows for every convergent series ~ u v , which m eans that v from the fact that an(a ) can be wri t ten as th e tran sforma tion (4) is regular. As an example, let us consider th e serics transfor _ 1 pK(z ,a) d _ j _ "( ~ 'Y v an( a ) - - , --n- z - -e L...J ,- (17) m ation 27r ~ Z v= ! V . 1 n U,, =- 1 ~ Pn - l ,v-1'U v (13) 11 .• = 1 where where Lbe PI/, a rc Lh e coeffi cicnts of the polynomial 'Y =-log (I - a). p ,,(z) =z(z+ 1) . . . (z+n - 1) (11 = 1,2, .. ,), (14) To check (iii) wc obscrve tha t This transformation was in troduced by Lototsky [3] and will be call ed the L-transformation . If th e 3 and therefore Assume (as we did in sec. 2) that the radius of con vergence of the series (21) is positive; in other words, that - ro- 1 lim ;;Iun l=-;-< 00. (22) n~co 1 Now it follows from theorem 1 that the transforma Since "jiJ--'7 00 it follows from (22) that condition tion (13) maps a convergent series into an A-sum (18) is automatically fulfilled. mabIe series with the same sum. In this special case it even can be shown that the The coefficients Un are represented by the integrals series "'22 Un is convergent itself. From (13) we get n (23) '" 1 n 1 1 L. a,,,=, :z= Pn-1Pv- l= , Pn-l(I) = - , v= 1 n. v= 1 n. n where the integral is taken around a certain circle Izl=R (R> r). From (23) it follows that and thus condition (12) is fulfilled which assures the convergence of the series "'22 U n. This means that n n "'22'" ~U t n=-1 . ~ ](z) "'22'" -znt dz=-.1 ~ j (z)eztdz, the L-method is regular. n=O n! 27r'l, n=O n! 27r'l, 3 . Connections Between the Lototsky and and therefore, using (19) and (20), the Borel Summabilities B ( t )=~ rt rF. ertz-l)](z)dzdr. (24) As men tioned before, thcre is a close relationship 27r'l, Jo J between the Lototsky and the Borel-integral methods Since the integrand is regular we can interchange the of summation. Let "'22u. be a (not necessarily con- • two integrations and integrating with respect to r we vergent) infinite series such that obtain 1 ~ etCz - 1)_1 B (t) = 2-----; 1 j(z)dz. (25) 7r'l, z- < 1-1m· ~. l-=un1 0. (18) n-->'" n! I Thus the BI-sum of the senes "'221( v can be repre I Then the sum sented as the limit F(t)= tV . 1 ~ et (z-1)_ 1 ~ 11 vT' (19) BI ( "'22 11 . ) = hm?"", - 1 ](z)dz. (26) 1= 0 V. v 1.-4 ID .•;. ./7r~ Z defines an entire function of t. The series "'22u. is Comparing transformations (16) and (25), we obtain • called summable with respect to the Borel integral the following relation between functions (a) and method (BI-summable) if the improper integral B(t) : ¢(a) = B (-log(l- a)) (2 7) which can also be written as exists. In this case we write (28) From this relation it follows immediately that the existence of one of the limits We are going to show that the function lim ( a) or lim B (t) a-->l-O t-->", (20) implies the existence of the other, and these limits coincide. This means in terms of series: Under the hypothesis (22) AL- and the BI-summability are can b e represented as a certain integral transform equivalent. of the function Here the question arises whether this equivalence still holds without condition (22 ). We are going '" 11 to show that the implication BI--'7AL is true in l' f(z) = "'22 --.;. (2 1) I v=1 z general but not conversely. 4 We first prove It has to be shown that the numbers an. coincide Ll~ MMA 1: Let u , be an injinite series such that the with the numbers Pn. defined by (14). FU'st of all function we have a".= O for v> n since the functions F(t)=£. u"+! t" [log (1- 0' )]" 1=0 v 0' 71+ 1 is regular in a certain neighborhood of t= O and define are regular at 0' = 0 provided that v~n+1. We now the functions B(t) and cf>(a ) by define the polynomials an(z) by n B(t)= .r F(r)e-Tdr an(z) = L:.: an,z" ,=1 and and show that cf>(a )=B(- log (1 - 0' )) . a,,( - k )=O (k = O, I, ... ,n-l ). Then the coefficients Un of the expans'ion In fac t we have for ever.\" in Leger ! (_ I )= ~lcV~ [IOg(l - a ) l ' l , a " Ie L..J, n+ 1 (i a are the L-transforms of u ,. n . .=1 V. a PROOF: The coeffi cients Un arc defined by B (-log ( l - a))= ~ Una n . n If now k ~ n - 1 Lhere is 110 residue at 0' = 0 and we get DifferenLiaLing this relation we get an(- lc )=O (1c = 0,1, ... ,11-1) . (3 2) 1 B'(- log (1 - a )) -I -= ~nUnan - l, (2 9) - a n From here it follows, since the polynomial an(z) has Lhe degree n, and hence, observing that a,,(z) = /.. z(z+ 1) . .. (z+n - l ) (33) B'(t)=F(t)e- t and therefore wllCre A is a consLant. Substituling Z= 1 in (33) we get B'(-log(l - a))= (l - a)F(- log (I - a)), we conclude that and on the other hand we have '" (-1)v l ~ nUnan- = ~ - -, - [log (l - a)]"uv+!. n 1 '" 1 '" a ,,(z)=z(z+ 1) ... (z+n- l )= Pn(z) U ,,=-, ~ a"_1 vUv+l =, L, an-I v-1u., (3 0 ) n. 1= 0 n. 1= 1 and comparing the coefficients wh ere the coefficien Ls an, are defmed by _(_ I )"n !~ . [log (I - a)]" l . n+1 GO' (31) a"v- V,. a which proves lemma 1. 5 THEOREM 2. 7 A BI-summable series is also AL- and the sequence Un by summable and the two sums coincide: (n= I,2, ... ). (36) Then we have PROOF: By hypo thesis the series and therefore this series is convergent, only for F(t)= i= UU ~I t' 1'=0 v. Itl defines an entire function. H ence the function To obtain the L-transform of sequence (36) we apply lemma 1. The function B(t) in our case is given by t e-rdT B (t ) = --=loo'(2-e- t ) i o 2-e-r b , is entire too, and therefore the function and from here we get cf> (a)= B( - log (l - a)) (34) cf>(a) = B (-log (I-a» = log (1 + a) , is regular for la l is convergent for la l< 1. Furthermore, we obtain ~ Uv+ l t' v v! from (34) and (35) has an infinite radius of convergence. It can b e ~ Unan= B (-log (l -a», shown however that this radius can not be less than n=1 log 2. To this purpose we prove LEMMA 2: Let ~u v be an infinite series and ~ Un its and hence, if a tends to 1 on the real axis, v n L-transjorm. A ssume that the series This m eans that the series ~u v IS AL-summable is convergent in a catain circle la ! is convergent in a certain circle Itl Conversely, one cannot conclude that an AL (3 7) s ummable (even an L-summable) series is also holds. BI-summable, as the following example shows: PROOF: By hy pothesis the function Let the function cp(t) be defined by (38) 1 cp( t) = 2-----=1 -e is regular in a certain circle Itl< p' and therefore the same is true for the function 1 Different proofs of theorems 2 to 4 are given in a mimeographed report by R. P . Agnew (U.S.A .F. Contract 18 (600-685» . (39) 6 I l ~ where B' indicates Lhe derivaLive of B. holds. PROOF: Since the series :Z::::; u, is AL-summable the From (38) and (3 9) it follows that fun ction F(t) = cp' ( l -et) .~ (40) Expanding the function F(t) into a power series, is regular for lal<1. Hence the function B (t) defin ed by F(t)= ~c . t ', v we obtain from (40) must be regular in the domain D , which IS deter mined by the inequality cp' (l -e- t )= :Z::::;c.t', v (45) and if we in troduce and the sam e holds for the function · as a new variable, F (t) = B' (t )e-t • cp' (a) = :Z::::; ( - l ) 'c v ~o g( I -a)]' . (41) v Writing t= x+iy, we get On the other hand we have II-e-t I2= 1+ 2e- x Ct e-x-cos y) q/(a) = :Z::::; nUnan- I. (42) n and hence inequality (45) is equivalen t to From (41 ) and (42) it follows that :z::::; nUna,,-I= :Z::::; (- I )'c, [log (I - a)]', This means that the point l= x+ iy lies in the n "interior" of the curve and from here we obtain x=-log (2 cos y). (46) y 1 = - j:Z::::; P n - J ,vic,. n. v This equation s tates that the Un arc the L transforms of the sequence v!c,. On the otJlCl' --~-----+----~------~x hand, they are by definition the L-transforms of u" and hence we conclude Therefore the expansion of F(t) becomes FIGURE 1. R elations between summation methods and integral transformations. F ( t ) = ~ U' ~I t' (43) v= o v. Thus the function B Ct) can be expanded into a power series within the largest circle Itl= p with the and this series must be convergent within the center t= O that is contained in the domain D . circle It I< p'. This is the circle R elation (3 7) now follows from (43), (39), and (38) . Itl=log 2. From lemma 2 we can derive Lhe following theorem, because this circle to uches curve (46) in the point which is proved by Agnew [2] in a different way. t= - log 2, it is sufficient to show that the radius of THEOREM. For the terms of an AL-summable curvature of the curve (46) is not less than log 2. series :Z::::;uv the inequality The curvature of this curve is given by v 1 lim n Il u nl ~ _I _ (44) -=--, 1l--t'" -V n! - log 2 P cos Y 508986- 59- 2 7 and hence we have where a (a 7'" 0) is an arbitrary complex number. In this case we obtain Thus the expansion of Fet) into a power senes must be convergent for and thus, ( "' ae-t log 2. ( ) Itl< BJ* ~ U v = Jo a+ t clt. On the other band, using lemma 2 this expansion is This integral converges for every number a which given by is not real and negative and therefore series (47 ) is BI*-summable for all those values of a. F (t)="L,UV+ l tV. The next example shows that a BI*-summable v v! series is not necessarily AL-summable (and hence not L-summable). Let the function cf>(t) be defined by From here it follows that (48) 1-·Jm V -IUnl <_ --1 1 7l-->'" n! -10g2 where a is a real number such that a> I, and let the sequence Un be defined by which proves theorem 3. It follows from series (36) that the bound l/log 2 can (n= 1,2, . . . ). ( 49) not be improved. The BI*-methocl. The application of the BI Then the series method provides that the series "L, U v , v (v + l)! t "L, U V ±! t' v v! is convergent in the circle converges in the whole t-plane. Frequently the Itl< log a case occurs that the radius of convergence of this series is positive and finite such that this series de and represents the function cf>(t). This function is fines a regular function F(t) within a certain circle defined along the whole positive t-axis and the integral ItI < po If this function can be analytically continued along the positive t-axis one can form the improper 1 integral a is convergent. Thus the series (49) IS BI*-sum mabIe. In case this integral is convergent, the series is said In order to obtain the L-transform of (49 ) accord to beBI*-summable [2, sec. 11] and its value is called ing to lemma 2 we have to form the function t.he BI*-sum of "L,uv, v t 1 1 B(t)= cf>(T) e-Tdr=- - --_-t io a- I a-e B J* (~Uv ) = i 00 F(t)e-tclt. and then substitute Obviously a BI-summable series is also BI*-sum t= -log (I - a) . mabIe to the same value. It can be shown that AL-summability implies This gives BI*-summability, but we do not prove this here since BI*-summability will turn out to be equivalent 1 1 cf>(a) to a generalization of AL-summability and the above a- I a- l + a statement will follow immediately from t.here. To demonstrate the BI*-method by an example, The function cf>(a ) so obtained has a pole at a = l-a let U n be defined by and hence the series"L,Una n is only convergent in the circle 7l _( 1)" n! U n+ l - - n (n = O, 1,2, ...) (47) a lal< a- l, 8 wh ich is smaller than the unit circle if a< 2. Therc This means that the sen es L:uv is AL*-summable fore, the AL-method can not be applied to series (49). The AL*-summability. As the last example shows and the relation the BI*-method is more powerful than the AL method. However the AL-method can be extended to a method AL* in a similar way as BI has been holds. extended to BI*. On the other hand, if th e serie L:uv is AL*-sum- Let L:uv be an infinite series such that the power v v mable, the function sen es ct> (a)=L: Un a" 11 is convergent within a circle lal< p and assume that IS regular within a circle lal< p. H ence the en (' the function cp(a) can be analy tically continued along the interval O~u < 1 (a = u +iv). If the F (t)=L: UV+ 1 tV limit v v! lim cp (a) (a real) a -;i-O according to lemma 2 is co nvergent within a circle ItI < p' , and th e relation exists we shall call the seri es L:uvAL*-snmmable and v F(t) = ct>' (l -e-')et (51) write holds. Since the function ct>(a) by hypoth esis can be AL* (L:uv) = lim cp (a). v a-;i - O analytically continll ed along the real unit interval the SAme holds for F(t) along the posi tive t-axis and AL*-summability obviously implies AL-summabil relation (51) hoMs for every positive t. From there ity. we obtain '" F(t)e-tdt= lim ct> (a) THEOREM 4. Th e AL*-summabihty is equivalent fo a-;J-O to the BI*-summab'iiity. which shows t}lat the series L:uv is BI*-summable. v PROOF: Assume first ('.] 1(' se ri es :L:uv JS BI*-sum- n Altogether we have the following relations between mable. Then the function the Lototsky and the Borel methods, where all the simple arrows are not invertable: F(t)=L: 1J v+ ~ tV v v! L -i>AL-i>AL* t 1i BI BI* i~ regular within a circle Itl< p. Let now the func tlOns B(t) and cp (a) be given b.v Series (36) shows that L does not imply BT, but it is not Imown to the author whether the implication B (t)= i t FH e- TdT BI -i>L holds. and 3. Application to Analytic Continuation cp(a)= B (- log (I - a)) . (50) Let g( z) be 31) analytic function regular in a do main D which contains the origin and let Then cp (a) is regular within a circle lal< p' and therefore It can be expanded into a power series g (z)= ~ U , Z·-l (52) v= 1 be the expansion of g( z) around the point z=O. It According to lemma 1 the coeffi cients Un are the is a known result that the series (52) is BI-summable g(z) L-transfonns of the U V ' By hypothesis, the function to the valu e at every interior point of the Borcl F (l) can be analytically continucd along the positive polygon 7r of g(z) and not BI-summable at every t-axis and therefore the same is true for the function exterior point. Making use of th e relationship B (t) . Hence eq (5 0) defmes an analvtic continua between the BI- and th e L-summability the same tion of the function ct>(a ) along the real unit interval can be shown for the L-summabili ty of series (52). and we have In order to do so we first observe that lim ct>(a)= lim B (t). lim ;/ I u"z~ 11=lzollim ../1UJ 9 and h ence the BI-summability is equivalent to AL this relation becomes summability (sec. 3). From this remark and the implication L -»AL it follows that series (52) can not b e L-summable at an exterior point of 7r. L et now Zo be an in terior point of 7r. Then the 1 series L: u vzg- is BI-summable and h ence AL-sum- 2 v = M lw(w+ 1) . . . (w+ n) I.1 n 1·lnw- 21 . mabIe. Thus the relation n!n w (w+ n - l )(w+n) (56) lim (L: U ,,(zo)an) = g(zo) (53) a-71-0 n If n tends to 00 the first factor approaches r (w) and therefore stays bounded. The same is true holds where Un (zo) denotes th e L-transform of the for the second term and therefore it is sufficient sequence uvzo-1. to show that the series It remains to show that the series ~U n(ZO) IS con- n vel'gent; then the relation (57) L: Un(zo) = g(zo) is uniformly convergent on c. The circle c IS n represented by the equation will follow from (53) and the Abelian limit theorem. ,liT e arc go in g to prove the absolu te convcrgence of the above series . For that we start ou t from the relations 1 11-1 From here it follows that U ,,(zo) = , L: Pn- l v U v+ l zg . (54) n. v=1 Since Zo is an interior point of the polygon 7r the closed circle and for the real part I z-~~2 I =< ~2 R (w)= 2(1 + p cos 0) ::;; _ 2_ . is contained in D , and therefore a number p>1 can 1+ 2p cos 0+ p2 -1+ p be chosen so close to 1 that the circle Therefore we have c: - 2p R (w- 2) ~ l + p= -1-8, still is contained in D. [2 , Chap. VIII.] Hence the coefficients U v can be represented by and hence the inequalities UV= 2~~g( :) dz . (55) 7r'/,J z hold, which imply the uniform convergence of series From (54) and (55) we obtain (57) on c. Now the absolute convergence of the series L:Un follows from (56). n 5 . Connections With the Euler-Knopp Method and from there, if 111 denotes the maxImum of g(z ) on the circle c The Euler-Knopp transform of a sequence u, z ' (EK-transform) is defined to be the sequence (58) If we write for abbreviation where K is a complex parameter. w=-,Zo This transformation can be derived from a lineal' Z transformation of the w-plane which has the fixed 10 point w= 1 and maps w= 0 into w= co. The most alld making usc of the iden ti ty genera 1 transformation of this kind is given by (_l )n-"= (n- l), l - a+ (-V) Z= -- a (59) n- v v - J w we finally get where a is a complex number. Now let )n- v U n= ~ (nv- l)l K" \' 1-x U •• f(z)= ~ :: (60) bf' an analytic function which is regular for Izl>l. This equation shows that t l1(\ coefficients U n are the This implies that the function EK-transforms of tlto coefficiell ts U v' From the relationslup betweell t lte EK-transJonna l - a ) tion and t lte lineal' mapping (59) it can easil y be f* (w) = f ( ----:-+a (6 1) derived that the ET\.-transfol'mation calln ot be regul ar unless K is r eal a nd O 11- a + aw l>lwl· l ' f( z)= ~ -'! (63) v Zv Since this domain co ntains the point w= O the functionf*(w) can be expa llded in to a power sf'ries has a sin g ul arity at a givell point 2, (lz, l= l , z !~1) . Theil the series ~ Un by hypothesis is also co nvergent f*(w) = ~ U nwn. " n nnd hell ce the functlOIl The coefficients U n are given by I -.a ) f '";w) = j ( --:;- +a. (64) Un= 2~ rh f~ , ~) dw. 7r ~ j W is regular [01' Iwl <' 1. Theref'or(' the fun ction fez) mllst be regular ill Lil e im age domain of the circle If we substitute here for the fun ction f*(w ) the Iwl 1 w ) " dw U n = 2--; ~ P( 1 n+ ! U v· 7r ~ v - a + aw w Therefore the poillt z, canDo t be contained in D and sin ce 2, was arbitrary on tbe unit circle this ll a; This integral can be evalu ated by the residue to hold for the whole circlf' Izl= l. Henco t he unit theorem. First we observe that circle has to be contained In the circle I z - a l ~ 11- a l = K" {I (K - i )w}-v + and this is possible only if a is real and a'::; 0 wllich m eans that K is real and O< K;;;: 1. - , where It is k.nown that an EK-summable series is also 1 K= ~ ' (62) AL-summable provided. that K is real and positive I - a [2 , sec. 8.] We shall glve here a proof of this fact using the integral repl'eSentatlOn of tbe L-trans From there we obtain formation. LeL ~u v be an EK-summable seri es and define the v functionj(2) by f (z )= ~ u v• v Z v 11 Then the function 6. Inversion Formula of the Lototsky Integral Transformation g(w) = f * (~) f (( I - a)w + a) The question arises whether the in tegral transfor mation has the expansion I ~ (l -a)I-Z- I cp (a) =~ 1 j(z)dz (65) ~7r ~ z- can be inverted. One cannot expect that to every The functionf(z) can be expressed by g(w) as given regular function cf> (a) (la l< 1) there exists a function fez) regular outside of a sufficiently large circle such that relation (65) holds. Provided such f (z)= g ( -z-a) . I - a a function exists, it can be represented as an integral transform of cp (a) at least in a half-plane. From this equation it follows that the L-integral THEOREM 5. Let j (z ) be a function regular for transform ofj(z) can be written as Izl>r such thatj(oo )=0, and let cp (a ) be its transform with respect to (65) . Then in the half-plane x>r the function j (z) can be represented as cf> (a) = 2. rho( l - a )I-Z-I f(z)dz 21f ~ j z- 1 j (z) = (z- l ).r cp (a)(1-a)'-2da. 1 (l - a)I-Z-l = - . ~ 9 ( --z-a) dz . 21f ~ z-1 I - a PROOF: Since.f(oo )=0, relation (65) can be written as Introducing the new variable 1 ~ (l_o:)I-Z cp (a) = -----;2 1 j (z) cl z. (66) ~ =z- a 7r ~ z- I - a Let zo=xo+iyo be a fixed point in the half-plane we obtain x> r. We choose for the path of integration in (65) a circle Izl=R, wherc and hence cf>(o:) can be considered as the transform of If (65) is multiplied by (l -a)Zo-2 and integrated with g(w) with respect to the kernel respect to 0: between 0 and t (O< t<1), we get 1 i t~ (1 )ZO -Z-I itcp (a) (l -a)'O-2cla= -2 . - a 1 f(z)d zdo:. o 7r ~ 0 z- It is easily verified that this kernel fulfills the concli Since the integrand is regular, the two integrations tions of theorem 1 provided th a t a is real an d 0 ~ a< 1. may be interchanged and we obtain integrating Therefore it fo llows that under this condition with respect to a lim cp (a)='L; Un = EK ('L;uv) a- >1-0 n v If t tends to 1, the e).1> resS10n provided that the series 'L; U" is convergent. But n that means that the series 'L;uv is AL-summable and (1- t) 'o-' v the rela tion tends to zero uniformly on 1 zl = R because of the inequality AL (~u v ) = EK (~uv ) holds. Hence we obtain In case K is not both real and positive it is shown by Agnew ([2], sec. 8) that EK-summability does 1 cf>o:( ) ( I -a ) ' 0-2 GO:l =-2'1 ~ ( f(z)dz1)( ).6()7 not imply AL-summability. io 1f~ z- Zo-Z 12 The r igh t-hancl side of thi equation can be evaluated correspond to each other by th e transformaLions by Lh e Cauchy integral formula since the function (65) and (68). fe z) is regular for Izl>r. Observing thatf(ro) = O, In particular, the function (70) posse ses the we get coefFi cien ts 1 rF, f (z) dz f (zo) (68) 27rij (z-l)(zo-z) = zo-( and hence we obtain from (72) From (67 ) and (68) it follows that f (zo) = ( I cf> (a) (1_ a) zO-2 da zo- l Jo Thus the function fe z) has the expansion and we obtain tb e formula (73) f(Z)=(Z - l ).r cf> (a ) (1- a )Z - 2da (x> r) (69 ) On the other hand this function is given by which is the inversion of (65). Ie! Formula (69) can be used to obtain the inverse f( z) = Pk(Z) (74) matrix of th e L-transformation (13). For the particular functions Comparing the right-hand sid es of (73) and (74) we conclude that (k = O, 1 . . . ) (70) ~ q n- I k - I _ I _ we get from (6 9) n z" Pk(Z) and from there k! Ie! Z (z+ l ) .. (z+ Ie - 1) Pk(Z ) or if n - l is replaced b y n and Ie - I by Ie, This function is in fact regular outside of the circle (75) Izl=k- 1 and t.hu s for the special functions (70) the integral equation (65) docs have a solution that is regular outside of a certian circle. This formula corresponds to the representation (3 1) To obtain the inverse of the L-transformaL ion of the matrix P nv. If the integral in (75) is evaluated by the residue theorem we obtain t he formul a 1 U,,=, ~ P n-I v-1Uv, (71 ) n . v k V ,,- I qnk= ~ (-1)"-' (v- 1) !(lc -v)! we denote by fi n} the illverse of the matrix P nj. Then it follows from (71 ) that = _ 1__ ± (_ I )n-. (lc- 1) vn - I • (Ie - I )! v= 1 v- I (72 ) 7. References The functions [II R . P . Agnew, The Lototsky method for evaluation of series Mich. Math. J. 4 (1957) . [2) G. H . Hard.v, Dive rgent series (Oxford, E ngland , 1940). [3) A. V. LototskYi On a linear t ra nsfo rmation of seq L1 enccs and a nd seri es, vanov. Gos. P ed . Inst . Dc. F iz.-Mat. Nauki 4,61 (1953) (i n Russia n). VVASHINGTON, D.C. (Paper 63Bl- 1) 13