On the Convergence of the Summation Formulas Constructed by Using a Symbolic Operator Approach

Transcription

1 On the Convergence of the Summation Formulas Constructed by Using a Symbolic Operator Approach Tian-Xiao He 1, Leetsch C. Hsu 2, and Peter J.-S. Shiue 3 1 Department of Mathematics and Computer Science Illinois Wesleyan University Bloomington, IL , USA, the@iwu.edu 2 Department of Mathematics, Dalian University of Technology Dalian , P. R. China, ulizhi63@hotmail.com 3 Department of Mathematical Sciences, University of Nevada Las Vegas Las Vegas, NV , USA, shiue@unlv.nevada.edu Abstract This paper deals with the convergence of the summation of power series of the form Saf; b ) a k b fk)k, where 0 a < b, and {fk)} is a given sequence of numbers with k [a, b) or ft) a differentiable function defined on [a, b). Here the summation is found by using the symbolic operator approach shown in [4]. We will give a different type of the remainder of the summation formulas. The convergence of the corresponding power series will be determined consequently. Some eamples such as the generalized Euler s transformation series will also be given. In addition, we will compare the convergence of the given series transforms. AMS Subect Classification: 65B10, 39A70, 41A80, 05A15. The research of this author was partially supported by Applied Research Initiative Grant of UCCSN 1

2 2 Key Words and Phrases: symbolic summation operator, power series, generating function, Euler s series transform, Polya and Szegö identity. 1 Introduction In [4], we present a symbolic summation operator with its various epansions, and construct several summation formulas with estimable remainders for Saf; b ) a k b fk)k, with the aid of some classical interpolation series due to Newton, Gauss and Everett, respectively, where 0 a < b, and {fk)} is a given sequence of numbers with k [a, b) or ft) a differentiable function defined on [a, b). In order to discuss the convergence of the summation formulas, we will give a new type of remainders. We now start from the following notations. It is known that the symbolic operations difference), E displacement) and D derivative) play important roles in the Calculus of Finite Differences as well as in certain topics of Computational Methods. For various classical results, see e.g., Jordan [6], Milne-Thomson [8], etc. Certainly, the theoretical basis of the symbolic methods could be found within the theory of formal power series, in as much as all the symbolic epressions treated are epressible as power series in, E or D, and all the operations employed are ust the same as those applied to formal power series. For some easily accessible references on formal series, we recommend Bourbaki [1], Comtet [2] and Wilf [10]. Recall that the operators, E and D may be defined via the following relations: ft) ft + 1) ft), Eft) ft + 1), Dft) d dt ft). Using the number 1 as an identity operator, viz. 1ft) ft), one can observe that these operators satisfy the formal relations E 1 + e D, E 1 e D 1, D log1 + ). Powers of these operators are defined in the usual way. In particular, one may define for any real number, viz., E ft) ft + ).

3 3 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) 1 E) 1 f0). This shows that the symbolic operator 1 E) 1 with parameter can be applied to ft) at t 0) to yield a power series or a generating function GF) for {fk)}. We shall need several definitions as follows. Definition 1.1 The epression ft) C[a,b) m m 1) means that ft) is a real function continuous together with its mth derivative on [a, b). Definition 1.2 α k ) is called an Eulerian fraction and may be epressed in the form cf. Comtet [2]) α k ) A), 1), 1 ) k+1 where A k ) is the kth degree Eulerian polynomial having the epression A k ) k Ak, ), A 0 ) 1 1 with Ak, ) being known as Eulerian numbers, epressible as Ak, ) ) k + 1 1) i i) k, 1 k). i i0 Definition 1.3 δ is Sheppard central difference operator defined by the relation δft) f t + 1 2) f t 1 2), so that cf. Jordan [6]) δ E 1/2 /E 1/2, δ 2k 2k E k. Definition 1.4 A sequence { n } will be called a null sequence if for any given positive rational) number ɛ, there eists an integer n 0 such that for every n > n 0 n < ɛ. For null sequences we quote the following result from [7] see Theorem 4 in 43).

4 4 Lemma 1.5 Let { 0, 1, } be a null sequence and suppose the coefficients a n,l of the system A {a i, : 0 i; i 0, 1, 2,...} satisfy the two conditions: i) Every column contains a null sequence, i.e., for fied p 0, a n,p 0 when n. ii) There eists a constant K such that the sum of the absolute values of the terms in any row, i.e., for every n, the sum a n,0 + a n, a n,n < K. Then the sequence formed by the numbers n a n,0 0 + a n,1 1 + a n, a n,n n is also a null sequence. Obviously, Lemma 1.5 is a consequence of the Toeplitz Theorem cf. P.74 in [7]). Definition 1.6 For any real or comple series a k, the so-called Cauchy root is defined by r lim k a k 1/k. Clearly, a k converges absolutely whenever r < 1. In [4], we have shown in 3 that 1 E) 1 could be epanded into series in various ways to derive various symbolic operational formulas as well as summation formulas for k 0 fk)k. For the completeness of the paper, we now cite the result as follows. Proposition 1.7 Let {fk)} be a given sequence of real or comple) numbers, and let gt) be infinitely differentiable at t 0. Then we have formally fk) k fk) k k1 fk) k k1 gk) k k 1 ) k+1 k f0) 1.1) ) k+1 δ 2k f1) δ 2k f0) ) 1.2) ) k+1 1 δ 2k f0) δ 2k f 1) ) 1.3) α k ) D k g0) k! 1.4) where we always assume that 0 and 1.

5 5 Remark 1.1 We may write cf. [5] and [9]) the Eulerian fractions α k ) as α k ) k!sk, ), 1.5) 1 ) +1 0 where Sk, ) are Stirling numbers of the second kind. Substituting Eq. 1.5) into Eq. 1.4) and noting! Dk k Sk, ) e D 1) cf. k! [1] and [4]) yields gk) k k 0 0 0! k! Sk, ) 1 ) +1 α k ) D k g0) k! 1 ) +1 Dk g0) )! Sk, ) Dk g0) k! k 1 ) +1 g0), which is the series epansion 1.1). Hence, in this paper, we only discuss the convergence of 1.1)-1.3). We shall give a new type of remainder in Section 2 for each of the summation formulas shown in the series transforms in Proposition 1.7. In Section 3, we shall discuss the convergence of the summation formulas by using the established remainders. Some eamples such as the generalized Euler s transformation series will also be given. In addition, we will compare the convergence of the given series transforms. 2 Summation Formulas with Remainders In this section we will establish three summation formulas with remainders. Theorem 2.1 Let {fk)} be a given sequence of numbers real or comple). Then we have formally

6 6 fk) k n 1 k 1 ) k+1 k f0) + n l n fl). 2.1) 1 ) n l0 n 1 fk) k δ 2k f1) δ 2k f0) ) ) n + l δ 2n fl) 2.2) l1 n 1 fk) k 1 δ 2k f0) δ 2k f 1) ) ) n + l δ 2n fl). 2.3) k1 k1 where we always assume that 0. Proof. From Eq. 1.1) we obtain l0 1 E) 1 1 ) 1 1 ) 1 1 { n 1 ) l 1 ) 1 l + ) n } ) l0 1 n 1 ) l n n 1 ) l+1 l E l0 n 1 ) l n 1 ) l+1 l + l E l n. 1 l0 l0 ) 1 1 Since E l n f0) n E l f0) n fl), applying operator 1 E) 1 and the rightmost operator shown in the above equalities to ft) t0 yields Eq. 2.1). Similarly, we can derive formula 2.2) formally as follows. From Eq. 1.2), we have

7 7 k fk) k1 n 1 δ 2k f1) δ 2k f0) ) k1 + 2k E k E )f0). kn Applying E 1 to the last series yields 2k E k E )f0) kn ) k+n+1 ) 2 k+n E )f0) E ) n+1 ) 2 n ) k ) 2 k E )f0) E E ) n+1 ) 2 n E f0) E ) 2 E ) n ) 2 n E ) f0) E 2 E ) n ) 2 n EE ) E E E 1) f0) ) 2 n ) 2 n E E 1 E f0) ) n k δ 2n fk), k1 which implies Eq. 2.2). Eq. 2.3) can be derived by using a similar argument. Remark 2.1 In Eq. 2.1), if we assume 1, then we have the following Euler s series transform

8 8 k fk) n 1 1) k 2 k+1 k f0) + 1)n 2 n 1) l n fl). 2.4) Hence, we may call 2.1) the generalized Euler s series transform, which can be used to accelerate the series convergence. We now use Theorem 2.1 to discuss the convergence of the transformation series 2.1)-2.3). Theorem 2.2 Let {fk)} be a given sequence of numbers in R or C), and let θ lim fk) 1/k. Then for any given with 0 we have the k convergent epressions 2.1), 2.2) and 2.3), provided that θ < 1/. Proof. Suppose that the condition θ < 1/ 0) is fulfilled, so that θ < 1. Hence the convergence of the series on the left-hand side of 2.1)-2.3) is obvious in accordance with the Cauchy s root test. To prove the convergence of the right-hand side of 2.1), it is sufficient to show that l0 l n fl) is absolutely convergent. Choose ρ > θ such that θ < ρ < 1. Thus, for large k we have fk) 1/k < ρ, i.e., fk) < ρ k. Consequently we have, for large l n fl) 1/l as l. Thus n 0 l0 ) 1/l n fl + ) ) ρ1 + ρ) n/l ρ n lim l n fl) 1/l ρ < 1, l 0 ) 1/l n )ρ l+ so that the series on the right-hand side of 2.1) is also convergent absolutely. The absolute convergence of the right-hand side series in 2.2) and 2.3) can be proved similarly. The only difference is to estimate δ 2n fl) 1/l which can be done as follows under the same condition θ < 1/.

9 9 δ 2n fl) 1/l 2n fl n) 1/l 2n 0 ) 1/l 2n fl n + ) ) ρ 1 n/l 1 + ρ) n/l ρ as l. It follows that 2n 0 lim l δ 2n fl) 1/l ρ < 1, l ) 1/l 2n )ρ l n+ which implies the absolute convergence of the right-hand side series of both 2.2) and 2.3). The argument in Theorem 2.2 applies to negative values of with 1. Thus for 1 we have the following corollary. Corollary 2.3 Let lim k fk) 1/k < 1. Then we have the convergent series 1) k fk) 1) k fk) 1) k fk) n 1 1) k k f0) 2 k+1 + 1)n 2 n 1) l n fl) 2.5) l0 n 1 1) k+1 δ2k f1) + δ 2k f0) 4 k+1 + 1)n 4 n 1) l δ 2n fl) 2.6) l1 n 1 1) k δ2k f0) + δ 2k f 1) 4 k+1 + 1)n 4 n 1) l δ 2n fl) 2.7) The condition shown in Theorem 2.2 can be replaced by the following weaker condition. l0

10 10 Theorem 2.4 Let {fk)} be a given sequence of numbers real or comple) such that fk)k is convergent. Then we have convergent epressions 2.1), 2.2), and 2.3) for every < 0. Proof. We write the remainder of epression 2.1) as follows. R n : n 1 ) n )n 1 ) n )n 1 ) n n 1 ) n l0 0 l n fl) l0 n n 1) n l 0 ) f + l) n ) n n 1) l l0 n ) n ) l fl) 0 l ) f + l) )n 1 ) n n ) n ), where l l fl) 0 n). Since l0 l fl) converges, is the term of a null sequence see Definition 1.4). To apply Lemma 1.5 here, we consider the coefficients ) a n, : )n n 1 ) n ). Hence, if is fied, for every [ 1, 0) we have a n, 0 as n because of ) a n, n n n < 1 ) n 1 ) n and 1/1 ) < 1, and for every < 1 we also have a n, 0 when n, seeing that it is ) ) n a n, n n < n 1 ) n 1 and /1 ) < 1. In addition, for every n and for every < 0 we have n a n, ) n 0 n ) n ) n 1. 0

11 11 Therefore, from Lemma 1.5, we find that R n is also the term of a null sequence, so the series on the right-hand side of 2.5) converges for every < 0. Using the same argument, we can show the convergence of the series on the right-hand side of Eqs. 2.6) and 2.7) for every < 0. Remark 2.2 If < 0, the convergence rate of the right-hand side series of either 2.2) or 2.3) is faster than the convergence rate of the right-hand series of 2.1) because the rate of the former two series are O/ ) n ) while the rate of the latter is O/1 )) n ), where / < /1 ) < 1. It is easy to see that the convergence of the series shown in 2.1)- 2.3) depends on both the property of f and the range of i.e., the convergence interval). From Theorems 2.2 and 2.4, we find that to ensure the convergence, more stringent requirements on f allow for weaker demands on the range of, and the reverse is also true. However, the epressions of 2.1)-2.3) show that the largest possible convergence interval for is < 1/2. To prove it, we need an alternate approach that will be shown in the net section. 3 An alternate approach for the convergence We will give other convergence conditions for series 2.1)-2.3). The series can be derived with the aid of the symbolic computation or more formally with the use of some identities, in which the largest possible convergence intervals for can be shown. Theorem 3.1 Let {fk)} be a given sequence of numbers real or comple). Then we have formally Eq. 2.1) for every < 1/2, Eq. 2.2) for every > or < 3 2 2, and Eq. 2.3) for every > or < and 0. Proof. From Eq. 1.1) we obtain k fk) n 1 k 1 ) k+1 k f0) + k 1 ) k+1 k f0) kn k 1 ) k+1 k f0).

12 12 Noting E 1, we write the last summation as kn k 1 ) k+1 k f0) n+k 1 ) n+k+1 n E 1) k f0)) k ) ) n+k k n E l 1) k l f0) 1 ) n+k+1 l n 1 ) n+1 n 1 ) n+1 n 1 ) n+1 l0 l0 k ) k k 1 ) k 1)k l n fl) l ) ) 1) l n ) k k fl) 1 ) k l kl ) k+l k + l 1) l n fl) 1 ) l l0 l0 ) ) 3.1). By using the well-known summation formula see e.g., [3], 1.3)) l0 ) k + l z k l we change the last series in 3.1) to be 1, z < 1, 1 z) k+1 n 1 ) n+1 l0 ) l n 1 fl) ) 1 ) 1 l+1, 1 for < 1/2, which is equivalent to the remainder shown in 2.1). We now derive formula 2.2). From Eq. 1.2), we have n 1 fk) k δ 2k f1) δ 2k f0) ) k1 + 2k E k E )f0). 3.2) k1 kn

13 13 Applying E 1 to the last series yields 2k E k E )f0) kn ) n+k+1 2n E n k E )E 1) 2k f0) ) n+k+1 2k ) 2k 2n E n E ) )E u k 1) u f0) u u0 ) n+k+1 k ) 2k 2n E n E ) )E l 1) l+k f0). l + k l k We split the last summation of the rightmost equality and obtain 2k E k E )f0) φ 1 f) + φ 2 f), kn where : φ 1 f) ) n+k+1 0 ) 2k 2n E n E ) )E l 1) l+k f0) l + k l k ) n+1 0 2k 1) l δ 2n E l E )f0) ) 1) k k + l l k l ) n+1 0 ) ) k l 2k 2l 1) l δ 2n E l E )f0) k l ) k and

14 14 : φ 2 f) ) n+k+1 k ) 2k 2n E n E ) )E l 1) l+k f0) l + k l1 ) n+1 2k 1) l δ 2n E l E )f0) ) 1) k k l l1 kl ) ) k+l 2k + 2l 1) l δ 2n fl + 1) fl)). k l1 We now apply the Polya and Szegö identity see e.g., [3] 1.120)) a + bk )z k k 1 )a b) to φ 1 f) and φ 2 f) with z /1 ) b, b 2, and a 2l and 2l, respectively, where z < b 1) b 1 /b b 1/4 or equivalently > or < Here, we need 1. However, this limitation will be omitted after we combine the resulting epressions of φ 1 f) and φ 2 f) later. Substituting the Polya and Szegö identity into the last epressions of φ 1 f) and φ 2 f) yields respectively ) k and φ 1 f) ) n+1 0 ) l 1) l δ 2n 1 ) 2l+1 fl + 1) fl)) 1 + l ) n+1 1) l δ 2n 1 ) )l f l + 1) f l)) 1 + l0 φ 2 f) ) n+1 1) l δ 2n fl + 1) fl)) l1 ) n+1 1) l δ 2n fl + 1) fl)) l1 ) l l ) )l. 1 +

15 15 Therefore, 2k E k E )f0) φ 1 f) + φ 2 f) kn { n+1 l δ 2n f l + 1) f l)) 1 + )n+1 l0 } + l δ 2n fl + 1) fl)) l1 ) { n l δ 2n f l + 1) f l)) 1 2 l1 } + l δ 2n fl + 1) fl)) l0 ) { n l+1 δ 2n f l) 1 2 l0 } + l 1 δ 2n fl) l+1 δ 2n fl) l1 l0 l+1 δ 2n f l) ) n l δ 2n fl). 3.3) l1 The rightmost equality shows that the limitation 1 is no longer needed. Hence, we obtain 2.2), which holds for all that satisfies either > or < Similarly, we can derive 2.3) and it completes the proof of Theorem 3.1. To give a compressed form of the remainders of the transform series shown as 2.1)-2.3), we need the following lemma see also in [4]). Lemma 3.2 Mean Value Theorem) Let n0 a n n with a n 0 be a convergent series for 0, 1). Suppose that φt) is a bounded continuous function 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 l1

16 16 Theorem 3.3 Let ft) be a bounded continuous function on, ). Then for < 1/2 fk) k n 1 k 1 ) k+1 k f0) + where ξ, ). For > or < n 1 ) n+1 n fξ), 3.4) n 1 fk) k δ 2k f1) δ 2k f0) ) k1 + n+1 n+1 δ2n fξ), 3.5) where ξ, ). Finally, for > or < and 0, we have n 1 fk) k 1 δ 2k f0) δ 2k f 1) ) k1 n + n+1 δ2n fξ), 3.6) where ξ, ). Proof. Clearly, 3.4)-3.6) are merely consequences of Theorem 3.1 and Lemma 3.2. Corollary 3.4 Let ft) be a uniformly bounded continuous function on, ). Then for < 1/2 fk) k For > or < and 1 k 1 ) k+1 k f0). 3.7) fk) k k1 δ 2k f1) δ 2k f0) ). 3.8)

17 17 Finally, for > or < and 1 and 0, we have fk) k k1 1 δ 2k f0) δ 2k f 1) ). 3.9) Proof. Taking limit n in Eqs. 3.4)-3.6) yields Eqs. 3.7)-3.9), respectively. Acknowledgments. The authors would like to thank the referee and the editor for their suggestions and help. References [1] N. Bourbaki, Algèbre, Chap. 4.5, Hermann, [2] L. Comtet, Advanced Combinatorics, Chap. 1,3, Dordrecht: Reidel, [3] H. W. Gould, Combinatorial Identities, Revised Edition, Morgantown, W. Va., [4] T. X. He, L. C. Hsu, P. J.-S. Shiue, and D. C. Torney, A symbolic operator approach to several summation formulas for power series, J. Comp. Appl. Math., ), [5] L. C. Hsu and P. J.-S. Shiue, On certain summation problems and generalizations of Eulerian polynomials and numbers, Discrete Mathematics, ), [6] Ch. Jordan, Calculus of Finite Differences, Chelsea, [7] K. Knopp, Theory and Application of Infinite Series, Hafner Publishing Comp., New York, [8] L. M. Milne-Thomson, The Calculus of Finite Differences, London, [9] X.-H. Wang and L. C. Hsu, A summation formula for power series using Eulerian fractions, Fibonacci Quarterly ), No. 1, [10] H. S. Wilf, Generating functionology, Acad. Press, New York, 1990.