A Symbolic Operator Approach to Several Summation Formulas for Power Series

Transcription

1 A Symbolic Operator Approach to Several Summation Formulas for Power Series T. X. He, L. C. Hsu 2, P. J.-S. Shiue 3, and D. C. Torney 4 Department of Mathematics and Computer Science Illinois Wesleyan University Bloomington, IL , USA 2 Department of Mathematics, Dalian University of Technology Dalian 6024, P. R. China 3 Department of Mathematical Sciences, University of Nevada Las Vegas Las Vegas, NV , USA 4 Theoretical Division, Los Alamos National Lab MS K70, Los Alamos, NM 87545, USA Abstract This paper deals with the summation problem of power series of the form S b af; ) a k b fk)k, where 0 a < b, and {fk)} is a given sequence of numbers with k [a, b) or ft) is a differentiable function defined on [a, b). We present a symbolic summation operator with its various epansions, and construct several summation formulas with estimable remainders for S b af; ), by the aid of some classical interpolation series due to Newton, Gauss and Everett, respectively. AMS Subject Classification: 65B0, 39A70, 4A80, 05A5. Key Words and Phrases: symbolic summation operator, power series, generating function, Eulerian fraction, Eulerian polynomial, Eulerian numbers, Newton s interpolation, Evertt s interpolation, Gauss interpolation. The research of this author was partially supported by Applied Research Initiative Grant of UCCSN

2 2 T. X. He, L. C. Hsu, P. J.-S. Shiue and D. C. Torney Introduction It is known that the symbolic operations difference), E displacement) and D derivative) play an important role in the Calculus of Finite Differences as well as in certain topics of Computational Methods. For various classical results, see, e.g., Jordan [7], Milne-Thomson [8], etc. Certainly, the theoretical basis of the symbolic methods could be found within the theory of formal power series, inasmuch as all the symbolic epressions treated are epressible as power series in, E or D, and all the operations employed are just the same as those applied to formal power series. For some easily accessible references on formal series, we may recommend Bourbaki [2], Comtet [3] and Wilf []. Recall that the operators, E and D may be defined via the following relations: ft) ft + ) ft), Eft) ft + ), Dft) d dt ft). Using the number as an identity operator, viz. ft) ft), one can observe that these operators satisfy the formal relations E + e D, E e D, D log + ). Powers of these operators are defined in the usual way. In particular, one may define for any real number, E ft) ft + ). Note that E k f0) [ E k ft) ] fk), so that any power series of t0 the form fk)k could be written symbolically as fk) k k E k f0) k 0 k 0 k 0E) k f0) E) f0). This shows that the symbolic operator E) with parameter can be applied to ft) at t 0) to yield a power series or a generating function for {fk)}. We shall show in 3 that E) could be epanded into series in various ways to derive various symbolic operational formulas as well as summation formulas for k 0 fk)k. Note that the closed form representation of series has been studied etensively. See, for eample, [9]

3 Symbolic operator approach to summation for power series 3 which presents a unified treatment of summation of series using function theoretic method. Some consequences of the summation formulas as well as the eamples will be shown in 4, can be useful for computational purpose, accelerating the series convergence. In Section 5, we shall give the remainders of the summation formulas. 2 Preliminaries We shall need several definitions as follows. Definition 2. The epression ft) C[a,b) m m ) means that ft) is a real function continuous together with its mth derivative on [a, b). Definition 2.2, 0,,, n represents a least interval containing and the numbers 0,,, n. Definition 2.3 α k ) is called an Eulerian fraction and may be epressed in the form cf. Comtet [3]) α k ) A k), ), ) k+ where A k ) is the kth degree Eulerian polynomial having the epression A k ) k Ak, j) j, A 0 ) j with the Ak, j) being known as Eulerian numbers, epressible as Ak, j) j ) k + ) i j i) k, j k). i i0 Definition 2.4 δ is Sheppard central difference operator defined by the relation δft) f t + 2) f t 2), so that cf. Jordan [7]) δ E /2 /E /2, δ 2k 2k E k. Moreover, in the sections 3 and 4, we will make use of several simple and well-known propositions which may be stated as lemmas as follows.

4 4 T. X. He, L. C. Hsu, P. J.-S. Shiue and D. C. Torney Lemma 2.5 There is a simple binomial identity mk ) m m k k, < ). ) k+ Lemma 2.6 Newton s symbolic epression for E is given by For f C n+ [0, ) E + ) ) k. k we have Newton s interpolation formula f) E f0) n where 0, ) and ξ, 0,,, n. ) ) k f0) + f n+) ξ), k n + Lemma 2.7 Euler s summation formula for the arithmetic-geometric series is given by j k j j0 A k) ) k+ α k), < ), where k is a positive integer, and α k ) is the Eulerian fraction. Lemma 2.8 For n we have Everett s symbolic epression cf. Jordan [7], 29). E For f C 2m, ) ) + k 2k 2k + E k + k 2k + ) 2k E k we have Everett s interpolation formula ). f) m ) + k δ 2k f) 2k + ) + m + f 2m) ξ), 2m ) ) + k δ 2k f0) 2k + where, ) and ξ, 0, ±,, ±m, m +.

5 Symbolic operator approach to summation for power series 5 Lemma 2.9 Gauss s symbolic epression for E is given by E + k 2k ) 2k E k + + k 2k + ) 2k+ E k+ ). For f C, ) 2m 29). we have Gauss interpolation formula cf. Jordan [7], f) m + k 2k ) 2k f k) + ) + m + f 2m) ξ), 2m ) ) + k 2k+ f k ) 2k + where, ) and ξ m, m ). Lemma 2.0 Mean Value Theorem) Let n0 a n n with a n 0 be a convergent series for 0, ). Suppose that φt) is a bounded continuous function of t on, ), and {t n } is a sequence of real numbers. Then there is a number ξ, ) such that a n φt n ) n φξ) a n n. n0 n0 3 Main Results We now state and prove the following proposition of various epansions of E). has four symbolic epan- Proposition 3. The operator E) sions, as follows.

6 6 T. X. He, L. C. Hsu, P. J.-S. Shiue and D. C. Torney E) k ) k+ k 3.) E) α k ) D k 3.2) k! ) k+ ) E) 2k + ) 2 2k 3.3) Ek E k ) k+ ) E) 2k + ) 2 E 2k 3.4) k E k+ where the condition is assumed, and moreover, 0 for 3.4). Proof. Here we present a proof in the sense of symbolic calculus, viz., every series epansion is considered as a formal series. Clearly 3.) may be derived as follows. E) + )) ) ) / )) k k ). k+ For proving 3.2) it suffices to make use of E e D and Lemma 2.7. Indeed we have E) e D ) k e kd k j0 kd) j j! ) k k j j0 D j j! j0 α j ) Dj j!. 3.3) and 3.4) can be justified in an entirely similar manner by using Lemma 2.5, Lemma 2.8 and Lemma 2.9, respectively. Indeed, 3.4) may be derived as follows.

7 Symbolic operator approach to summation for power series 7 E) E) j j { ) j + k ) j 2k 2k E + ) } j + k ) j 2k+ k 2k + E k+ j j { } k 2k ) 2k+ E + k+ 2k+ k ) 2k+2 E k+ ) k+ ) 2k ) 2 E + 2k+ k E k+ ) k+ ) 2k ) 2 E 2k. k E k+ Once 3.3) is derived by the aid of Lemma 2.5 and Lemma 2.8, it can also be verified by symbolic computations. In fact we have + RHS of 3.3) + E ) 2 2 ) 2 E ) 2 + ) k ) 2 k E ) ) 2 E E ) ) 2 2 E EE ) + ) 2 E E ) + E 2 E E) LHS of 3.3). Certainly 3.4) could also be verified in the like manner as above. Remark 3. Note that all the operators displayed on the right-hand sides of 3.) - 3.4) involve or D, so that they will yield finite epressions when they are applied to any polynomial ft) at t 0. In particular, we see that for the pth degree polynomial ft), 3.) gives a generating function GF ) in the form fk) k p k ) k+ k f0). 3.5)

8 8 T. X. He, L. C. Hsu, P. J.-S. Shiue and D. C. Torney Actually, this is a well-known formula and was mentioned in Jordan [7],. Moreover, an eact formula parallel to 3.5) may be obtained from 3.2), namely fk) k p α k ) D k f0). 3.6) k! Certainly, both 3.5)) and 3.6) may be used either as summation formulas for the power series fk)k with <, or as a tool for getting GF s for the sequence {fk)}. Remark 3.2 Observe that Euler s formula as given by Lemma 2.7 is a particular case of 3.6) with ft) t p p ). Obviously, Euler s formula may also be deduced from 3.5) by recalling the fact that cf. Hsu & Shiue [5]) α k ) k j j!sk, j) ), j+ j0 where Sk, j) are Stirling numbers of the second kind. Proposition 3.2 Let {fk)} be a given sequence of numbers real or comple), and let ht) be infinitely differentiable at t 0. Then we have formally fk) k hk) k fk) k k fk) k k k ) k+ k f0) 3.7) α k ) D k h0) k! 3.8) ) k+ δ 2k f) δ 2k f0) ) ) 2 3.9) ) k+ δ 2k f0) δ 2k f ) ) 3.0) ) 2 where we always assume that 0 and. Proof. Clearly 3.7) - 3.0) are merely consequences of 3.) - 3.4) by applying the operators to ft) or ht) at t 0.

9 Symbolic operator approach to summation for power series 9 As in the case of 3.5) - 3.6), we have a corollary form 3.9) - 3.0), namely Corollary 3.3 If ft) is a polynomial in t of degree p, then fk) k k fk) k k [p/2] ) k+ δ 2k f) δ 2k f0) ) 3.) ) 2 [p/2] ) k+ δ 2k f0) δ 2k f ) ) 3.2) ) 2 Certainly, 3.) - 3.2) may also be used as a rule for obtaining GF s of {fk)}. 4 Consequences of Proposition 3.2 and Eamples As observed in 3, any of the formulas 3.5), 3.6), 3.) and 3.2) solves generally the summation problem of power series fk)k in the case ft) is a polynomial. Thus for instance, a few summation formulas of the forms k + λ θ) p k D p k, α) k p k!sp, k, λ θ) k ) k+ 4.) p α k!sp, k, α θ) k ) k+ 4.2) as given in Hsu and Shiue [5] are just particular cases of 3.5) in which ft) t+λ θ) p and ft) D p t, α) are known as the generalized falling factorial and the Dickson polynomial, respectively, or more precisely t + λ θ) p Π p j0 t + λ jθ), p ), t + λ θ) 0, and [p/2] ) p p j D p t, α) α) j t p 2j, D 0 t, α) 2. p j j j0

10 0 T. X. He, L. C. Hsu, P. J.-S. Shiue and D. C. Torney Moreover, Sp, k, λ θ) denotes Howard s degenerate weighted Stirling numbers. For more in details, cf. [5] loc. cit.) Another important consequence of Proposition 3.2 is that 3.7), 3.9) and 3.0) with yield three series transforms respectively ) k fk) ) k fk) k ) k fk) k ) k 2 k+ k f0) 4.3) ) k 4 k+ δ 2k f) + δ 2k f0) ) 4.4) ) k 4 k+ δ 2k f0) + δ 2k f ) ). 4.5) Note that 4.3) is the well-known Euler series transform that can be used to convert a slowly convergent alternating series )k fk) with fk) 0 as k ) into rapidly convergent series. For instance, the series ln ) can be converted using 4.3) with fk) quickly convergent series of the form k+ k 0,, 2,...) into a ln ) Actually, the above epression can be derived by substituting k f0) k j0 ) k ) k j j + ) 4.8) j into 4.3). Thus,

11 Symbolic operator approach to summation for power series ln 2 ) k fk) k + )2 k+ k j0 ) k 2 k+ k j0 ) k + ) j j + ) k ) k j j + ) j k + )2 k+ k2. 4.9) k k Remark 4. Obviously, the convergence of the series shown in 4.7) with a rate of O/2 n ) is much faster than the convergence of the series in 4.6), which has the rate of O/n). For instance, to arrive the accuracy of the five digits of ln , we only need to sum the first 5 terms of the series in 4.7), while the partial sum of the first 40, 000 terms of the series in 4.6) is ) - 4.5) appear to be novel, and they could also be used to convert slowly convergent alternating series k )k fk) into quickly convergent ones if a definition for fk) 0 k 0,, 2,...) is introduced. A later work will give the comparison on the rate of the convergence of series 4.3)-4.5) for the positive decreasing functions. We now give some eamples of the summations shown in Proposition 3.2. Our first eample is for function f) / + ) 2. Similar to epression 4.8) we obtain k f0) k j0 ) k ) k j j + ) 2. j Substituting the above epression into 3.2) and noting the well-known identity cf. [4]) k ) k ) j j j j see [6]), we use the process similar to that in 4.9) and have k l l,

12 2 T. X. He, L. C. Hsu, P. J.-S. Shiue and D. C. Torney ) k fk) k k + )2 k+ k2 k k l k j0 l j0 2 k+ k j0 ) k + ) j j + l ) k ) j j j + ) 2 j + k k2 k k j lj + l)2 j+l l σ, l l ) k ) j j j where summation 360c) in Jolley [6] gives the first sum as π2 2 2 ln2 2 and easily seen to equal σ j l lj + l)2 j+l yielding 2 j l jl2 j+l 2 ln2 2 ) k k + ) 2 π ) Remark 4.2 Although formula 4.0) can be easily derived by using Fourier cosine epansion of 2, we give a different approach here by using formula 4.3) because it converts the series in 4.0) into the following quickly convergent series: π 2 2 ) k k + ) 2 k k2 k Hence, we can use the last series shown above to evaluate ζ2) as ζ2) π2 6 k2 k k k l k l l. l.

13 Symbolic operator approach to summation for power series 3 The sum of the first 3 terms of the last series gives.6449, the first 5 digits of π 2 /6, while the sum of the first 5, 000 terms of the series in 4.0) is only This eample shows that formula 4.3) can be used to convert an alternating series into quickly convergent ones. We now consider another eample generated by function f) gt)), where g : R R and f is defined on N {0}. Obviously, we have and for i 0, k f0) k j0 ) k gt)) j ) k j gt) ) k 4.) j δ 2k fi) 2k E k fi) 2k gt)) i k 2k ) 2k gt)) i k+j ) 2k j gt) ) 2k gt)) i k. 4.2) j j0 Hence, substituting 4.) into 3.7) yields gt)) k k gt) ) k gt) )k ) k+ gt). 4.3) Similarly, substituting 4.2) into 3.9), we obtain the following summation formula ) k+ gt) ) { 2k gt)) k gt)) k} ) 2 gt)gt) ) ) gt) ) 2 k+ gt) ) 2 gt) ) 2 gt)gt) ) gt) ) 2 gt) ) 2 gt) ) 2 gt) ) 2 gt)gt) ) gt) + 2 ) gt)) 2 + ) gt) gt). 4.4)

14 4 T. X. He, L. C. Hsu, P. J.-S. Shiue and D. C. Torney As eamples, we take gt) e it, with i, and gt) t. Thus, from 4.3) we have respectively and e itk k k ) k+ eit ) k e it 4.5) t) k By applying 4.4) for gt) e it and t we obtain and ) 2 ) 2 k ) gt) k+ )k t. 4.6) ) k+ e it ) 2k { e ik )t e ikt} eit e it 4.7) ) k+ t ) 2k { t k t k} t t, 4.8) respectively. We now illustrate 3.8) with h) gt)) with g : R R and gt) > 0. Hence, D k h0) ln gt)) k and from 3.8) and Definition 2.3 gt)) k k gt) α k ) ln gt)) k k! A k ) k! ) k+ ln gt))k. 4.9) Replacing ln gt) by t and t ), respectively, in Eq. 4.9) yields GF s cf. Section 6.5 in [3] and [0]) and α k ) tk k! e t 4.20)

15 Symbolic operator approach to summation for power series 5 A k ) tk k!. 4.2) et ) Some other GF s such as 5i) 5k) shown in Section 6.5 in [3] can be derived from Eq. 4.9). In addition, from Eq. 4.20), we can establish the recurrence relation for α k ) by multiplying both sides of the equation by e t ). The details can be found in [0]. Finally, we consider a special case of 4.9) by letting h) e it, i.e., gt) e it with t R, and in 4.9), we obtain e ikt ) k + e { i tan t } it 2 2 A k ) k!2 k+ it)k. Therefore, direct verification of the rightmost equality would be effected by the identity A k )z k tanh z, 4.22) k! k implying A k ) is the negative of the respective tangent coefficient [7]. Note that A k ) k Ak, j) ) j j k j!sk, j) 2) k j, 4.23) with Sk, j) denoting the Stirling number of the second kind cf. Formula [5l] in 6.5 of [3], [2] and [5]). Therefore, implementing eponential GFs, in z, on both sides of 4.23), 4.22) follows from j! kj Sk, j)zk /k! e z ) j cf. []). The numerical results in Remarks 4. and 4.2 are obtained by using mathematical package Mathematica. All of the sums, ecept for the few discussed in the eamples, can also be done with any mathematical package, for eamples Mathematica and Maple. 5 Summation Formulas with Remainders In this section we will establish four summation formulas with remainders whose forms are suggested by Lemmas 2.6, 2.8 and 2.9. j

16 6 T. X. He, L. C. Hsu, P. J.-S. Shiue and D. C. Torney Theorem 5. Let ft) C[0, ) m, m ), with bounded derivative f m) t) in [0, ), and let fk)k be convergent for <. Then for 0, ) we have N km fk) k m M k fm) N k fn) ) k ) k+ + ρ m, 5.) where the remainder ρ m has a form with ξ [0, ) as follows ρ m M f m) M + ξ) N f m) N + ξ) ) m. 5.2) ) m+ Proof. Let φt) φt, ) M ft + M) N ft + N), so that φt) C[0, ) m. Then by Lemma 2.6 and using the Mean-Value Theorem Lemma 2.0) with a n n m), we obtain N fk) k km { m m φn) n n0 ) } n ) n k φ0) n + φ m) ξ n ) n ξ n n, 0,, 2,, m ) k m n0 n0 m ) n ) n k φ0) ) n + φ m) ξ) n k m n0 n0 k k φ0) ) + m k+ φm) ξ) ) m+ RHS of 5.) with ρ m being given by 5.2). Note that the RHS of 5.) without ρ m may be regarded as a rational approimation to the series on the LHS. In particular, if N k fn) 0 N, 0 k m ), then 5.) - 5.2) reduce to fk) k km m k m M k fm) ) + k+ M f m) M + ξ) ). m+ 5.3)

17 Symbolic operator approach to summation for power series 7 Theorem 5.2 Under the same condition of Theorem 5., we have N km fk) k m where the remainder is given by M f k) M) N f k) N) ) α k ) k! + ρ m, 5.4) ρ m M f m) M + ξ) N f m) N + ξ) ) α m ). 5.5) m! Proof. Denote φt) M ft + M) N ft + N) so that φt) C m [0, ). Clearly, by using Taylor s epansion with Lagrange s remainder, we have N fk) k km m n0 m φn) n n0 ) k! φk) 0)n k n + n0 ) k! φk) 0) n k n + S 2. n0 m! φm) ξ n )n m n, 0 < ξ n < n) Here we can apply Lemma 2.0 to the series S 2, thus obtaining S 2 ) m! φm) ξ) n m n n0 m! φm) ξ)α m ) ρ m. 0 < ξ < ) Hence, in accordance with Lemma 2.7, we have 5.4) and 5.5). Remark 5. Theorem 5.2 with epressions 5.4) - 5.5) is of similar nature as that of Theorems and 2 in Wang & Hsu [0]. However, our present result appears to be a little more restrictive since we have assumed here the condition 0 < < and the convergence of fk)k for <. In what follows we shall give formulas using the central difference operators δ 2k 2k /E k which appear to be more available for numerical computations.

18 8 T. X. He, L. C. Hsu, P. J.-S. Shiue and D. C. Torney Theorem 5.3 Let ft) C, ) 2m with bounded derivative f 2m) t) in, ) and let fk)k be convergent for <. Then for 0, ), we have N km fk) k m δ 2k φ) δ 2k φ0) ) ) k+ + ρ ) 2 m, 5.6) where δ 2k φt) M δ 2k ft + M) N δ 2k ft + N) and ρ m is given by the following with ξ [ m, ) ρ m M f 2m) M + ξ) N f 2m) N + ξ) ) m+. 5.7) ) 2m+ Proof. Denote φt) M ft+m) N ft+n), so that φt) C 2m, ). Let us now make use of Everett s formula Lemma 2.8) for φt) at t n, φn) m [ ) ) ] n + k n + k δ 2k φ) δ 2k φ0) 2k + 2k + ) n + m + φ 2m) ξ n ), 2m where ξ n n, 0, ±, ±2,, ±m, m +. Clearly we have

19 Symbolic operator approach to summation for power series 9 N km m m fk) k φn) n, 0 < < ) n0 n0 ) ] n + k δ 2k φ0) n 2k + [ ) n + k δ 2k φ) 2k + n0 ) n + m + φ 2m) ξ n ) n 2m n0 m ) n + k δ 2k φ) ) n 2k + n0 m ) n + k δ 2k φ0) ) n + ρ m 2k + δ 2k k+ m φ) ) δ 2k k+2 φ0) 2k+2 ) + ρ 2k+2 m m δ 2k φ) δ 2k φ0) ) ) 2 ) k+ + ρ m. Here an application of Lemma 2.0 to the series representation of ρ m yields ) n + m ρ m φ 2m) ξ) ) n φ 2m) m+ ξ) 2m ), 2m+ n0 where ξ [ m, ). Hence the theorem is proved. Theorem 5.4 Under the same condition of Theorem 5.3 we have N km fk) k m δ 2k φ0) δ 2k φ ) ) ) k+ + ρ ) 2 m, 5.8) where 0 and δ 2k φt) M δ 2k ft + M) N δ 2k ft + N), and

20 20 T. X. He, L. C. Hsu, P. J.-S. Shiue and D. C. Torney ρ m M f 2m) M + ξ) N f 2m) N + ξ) ) m. 5.9) ) 2m+ Proof. As before, denote φt) M ft + M) N ft + N) and let 0, ). Using Gauss interpolation formula with remainder Lemma 2.9) for φt) at t n, we get as in the case of proving Theorem 5.3 the following epressions N km m fk) k φn) n n0 ) n + k 2k φ k) ) n 2k n0 2k+ φ k ) m + + n + m 2m n0 m n0 ) φ 2m) ξ n ) n ) n + k ) n 2k + 2k k m φ k) ) + 2k+ k+ φ k ) 2k+ ) + ρ 2k+2 m m 2k φ k) + [ 2k φ k) 2k φ k ) ] ) ) k+ + ρ ) 2 m m 2k φ k) 2k φ k ) ) ) 2 ) k+ + ρ m. Finally, an application of Lemma 2.0 to the series epression of ρ m gives ρ m φ 2m) ξ) ) n + m n φ 2m) m ξ) 2m ), 2m+ n0 where ξ, ). Hence Theorem 5.4 is proved.

21 Symbolic operator approach to summation for power series 2 Remark 5.2 The uniform boundedness conditions for f m) t) in [0, ) as well as for f 2m) t) in, ) imply that N f m) N + ξ) 0 and N f 2m) N + ξ) 0 as N and 0 < <. Thus, if in addition, N f k) N) on), N, 0 k m ), then 5.4) - 5.5) yield fk) k km m M f k) M) α k) k! + M f m) M + ξ) α m). 5.0) m! Similar consequences from Theorems 5.3 and 5.4 may also be deduced by providing additional conditions such as N δ 2k fn) 0 N, 0 < < ). Acknowledgments. The authors would like to thank the referees and the editor for their suggestions and help. References [] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, U. S. Government Printing Office, 964; 2..4 B). [2] N. Bourbaki, Algèbre, Chap. 4.5, Hermann, 959. [3] L. Comtet, Advanced Combinatorics, Dordrecht: Reidel, 974 Chapters, 3). [4] H. W. Gould, Combinatorial Identities, Revised Edition, Morgantown, W. Va., 972. [5] L. C. Hsu and P. J.-S. Shiue, On certain summation problems and generalizations of Eulerian polynomials and numbers, Discrete Mathematics, ), [6] L. B. W. Jolley, Summation of series, 2d Revised Edition, Dover Publications, New York, 96. [7] Ch. Jordan, Calculus of Finite Differences, Chelsea, 965. [8] L. M. Milne-Thomson, The Calculus of Finite Differences, London, 933.

22 22 T. X. He, L. C. Hsu, P. J.-S. Shiue and D. C. Torney [9] A. Sofo, Computational Techniques for the Summation of Series, Kluwer Acad., New york, [0] X.-H. Wang and L. C. Hsu, A summation formula for power series using Eulerian fractions, Fibonacci Quarterly ), No., [] H. S. Wilf, Generatingfunctionology, Acad. Press, New York, 990.