Appendix A Euler-MacLaurin and Euler-Boole Formulas A.1 A Taylor Formula The classical Taylor formula Z Xm xk x .x t/m f .x/ D @kf .0/ C @mC1f .t/dt kŠ 0 mŠ kD0 xk can be generalized if we replace the polynomial kŠ by other polynomials (Viskov 1988; Bourbaki 1959). Definition If is a linear form on C0.R/ such that .1/ D 1,wedefinethe polynomials .Pn/ by: P0 D 1 @Pn D Pn1 , .Pn/ D 0 for n 1 P . / k The Generating Function k0 Pk x z We have formally X X X k k k @x. Pk.x/z / D Pk1.x/z D z. Pk.x/z / k0 k1 k0 © Springer International Publishing AG 2017 175 B. Candelpergher, Ramanujan Summation of Divergent Series, Lecture Notes in Mathematics 2185, DOI 10.1007/978-3-319-63630-6 176 A Euler-MacLaurin and Euler-Boole Formulas thus X k xz Pk.x/z D C.z/e k0 To evaluate C.z/ we use the notation x for and by definition of .Pn/ we can write X X k k x. Pk.x/z / D x.Pk.x//z D 1 k0 k0 X k xz xz x. Pk.x/z / D x.C.z/e / D C.z/x.e / k0 1 this gives C.z/ D xz . Thus the generating function of the sequence .Pn/ is x.e / X n xz Pn.x/z D e =M.z/ n xz where the function M is defined by M.z/ D x.e /: Examples P xn n xz . / .0/; . / . / 1; . / (1) f D f Pn x D nŠ , M z D n0 Pn x z D e R 1 R 1 Bn.x/ zx 1 z . / . / . / . / . 1/ (2) f D 0 f t dt , Pn x D nŠ , M z D 0 e dx D z e X B .x/ zexz n zn D nŠ ez 1 n0 The Bn.x/ are the Bernoulli polynomials and the Bn D Bn.0/ the Bernoulli numbers. With the generating function we verify that B0 D 1, B1 D1=2, n B2nC1 D 0 if n 1, Bn.1 x/ D .1/ Bn.x/. / . / 1 . .0/ .1// . / En x (3) f D 2 f C f , Pn x D nŠ X E .x/ 2exz n zn D nŠ ez C 1 n0 The En.x/ are the Euler polynomials and we set En D En.0/. With the generating function we verify that E0 D 1; E1 D1=2 if n 1, n En.1 x/ D .1/ En.x/. A.2 The Euler-MacLaurin Formula 177 The Taylor Formula Let f be a function in C1.R/,thenwehave Z x f .x/ D f . y/ C @P1.x C y t/@f .t/dt y and by integration by parts we get for every m 1 Xm k k f .x/ D f . y/ C .Pk.x/@ f . y/ Pk. y/@ f .x// kD1 Z x mC1 C Pm.x C y t/@ f .t/dt y Applying to this function as a function of y gives a general Taylor formula: for every m 0 Z Xm x k mC1 f .x/ D y.@ f . y//Pk.x/ C y. Pm.x C y t/@ f .t/dt/ kD0 y A.2 The Euler-MacLaurin Formula We can transform the Taylor formula to get a summation formula. Taking x D 0 we get Z Xm y k mC1 f .0/ D y.@ f . y//Pk.0/ y. Pm. y t/@ f .t/dt/ 0 kD0 R 1 In the case of W f 7! 0 f .t/dt we have Z Z Xm 1 y . / Bk k1 1 Bm y t mC1 f .0/ D @ f 0 . @ f .t/dt/dy kŠ 0 0 mŠ kD0 Replacing m by 2m and with B1 D1=2 and B2kC1 D 0,weget Z 1 1 Xm B2k 2k1 1 f .0/ D f .t/dt C . f .0/ f .1// C @ f 0 0 2 .2k/Š kD1 Z Z 1 y B2 . y t/ . m @2mC1f .t/dt/dy 0 0 .2m/Š 178 A Euler-MacLaurin and Euler-Boole Formulas The last integral can easily be evaluated by Fubini’s theorem, we get Z 1 1 Xm B2k 2k1 1 f .0/ D f .t/dt C . f .0/ f .1// C @ f 0 0 2 .2k/Š kD1 Z 1 B2 C1.t/ C m @2mC1f .t/dt 0 .2m C 1/Š Let j be a positive integer, by replacing f by x 7! f . j C x/ in the last formula, we have Z jC1 Xm 1 B2 C1 f . j/ D f .t/dt C . f . j/ f . j C 1// C k @2k1f j 2 .2k/Š j j kD1 Z jC1 b2 C1.t/ C m @2mC1f .t/dt j .2m C 1/Š where b2mC1.t/ D B2mC1.t Œt/. Summing these relations for j from 1 to n 1; we get for f 2 C1.0; 1Œ/ the Euler-MacLaurin formula Z n f .1/ C f .n/ f .1/ C :::C f .n/ D f .x/dx C 1 2 Xm B2 C k Œ@2k1f n .2k/Š 1 kD1 Z n b2 C1.x/ C m @2mC1f .x/dx 1 .2m C 1/Š A.3 The Euler-Boole Formula In the case of the Euler polynomials, the formula Z Xm y k mC1 f .0/ D y.@ f . y//Pk.0/ y. Pm. y t/@ f .t/dt/ 0 kD0 gives Z Xm 1 E 1 1 .1/mE .t/ f .0/ D .@kf .0/ C @kf .1// k m @mC1f .t/dt 2 kŠ 2 0 mŠ kD0 A.3 The Euler-Boole Formula 179 Let j be a positive integer, by replacing f by x 7! f . j C x/ in the last formula we obtain Xm 1 E f . j/ D .@kf . j/ C @kf . j C 1// k 2 kŠ kD0 Z 1 jC1 .1/mE m .t j/@mC1f .t/ dt 2 j mŠ Let’s define Œt m em.t/ D .1/ .1/ Em .t Œt/ we obtain by summation on j the Euler-Boole summation formula 1 Xm E f .1/ f .2/ C :::C .1/n1f .n/ D @kf .1/ k 2 kŠ kD0 .1/n1 Xm E C @kf .n C 1/ k 2 kŠ kD0 Z 1 nC1 1 mC1 C em .t/@ f .t/ dt 2 1 mŠ Appendix B Ramanujan’s Interpolation Formula and Carlson’s Theorem We give a proof of the following theorem. Carlson’s Theorem Let f be an analytic function in the half-plane Re.z/>d where 0<d <1. Let’s assume there exist a >0and b <such that j f .z/j Ä aebjzj for every z with Re.z/>d Then the condition f .n/ D 0 for n D 0;1;2;:::;implies f D 0: We prove this theorem by the use of an interpolation formula which is related to Ramanujan’s interpolation formula. First in Theorem 1 below we get an integral formula for the function XC1 g W x 7! f .n/.1/nxn nD0 which is Z 1 cCi1 g.x/ D f .u/xudu 2i ci1 sin.u/ Then in Theorem 2 we prove the interpolation formula Z sin.z/ C1 f .z/ D g.x/xz1dx 0 Since for the definition of the function g we only need to know the values f .n/; n D 0;1;2;::::, we see that this interpolation formula determines the function f in the half-plane Re.z/<d when we only know the values f .0/; f .1/; f .2/; : : :, thus we have a proof of Carlson’s theorem. © Springer International Publishing AG 2017 181 B. Candelpergher, Ramanujan Summation of Divergent Series, Lecture Notes in Mathematics 2185, DOI 10.1007/978-3-319-63630-6 182 B Ramanujan’s Interpolation Formula and Carlson’s Theorem Theorem 1 Let f be an analytic function in the half-plane Re.z/>dwhere 0<d <1. Let’s assume there exist a >0and b <such that j f .z/j Ä aebjzj for every z with Re.z/>d P . /. 1/n n Then the series n0 f n x is convergent in D.0; eb/ Dfx 2 C; jxj < ebg and defines an analytic function g in D.0; eb/. This function g has an analytic continuation in Sb DfjArg.z/j < bg which is defined by Z 1 cCi1 g W z 7! f .u/zudu 2i ci1 sin.u/ Proof For 0 Ä ˛<eb we have j f .n/.1/nxnj Ä aebn˛n P . /. 1/n n .0; ˛/ Thus the series Pn0 f n x is normally convergent in D and defines a C1 . /. 1/n n .0; b/ function g W z 7! nD0 f n z that is analytic in D e . For 0<x < eb and 0<c < d, the function u 7! f .u/xu is analytic in the half-plane Re.u/<d, and we consider the integral Z 1 f . u/xudu 2 . / i N sin u where N is the path iN -N-1/2 O c -iN . / u 0; 1; 2;:::; ;::: The function u 7! sin.u/ f u x has simple poles at n with Res. In/ D .1/nf .n/xn sin.u/ B Ramanujan’s Interpolation Formula and Carlson’s Theorem 183 thus we get Z 1 XN f .u/xudu D f .n/.1/nxn 2i sin.u/ N nD0 Lemma 1 When N !C1we have for 0<x < eb Z Z 1 1 cCi1 f .