B. Harris and L. Schoenfeld, Asymptotic expansion for the coefficients of analytic functions, Illinois J. Math. 12 (1968), 264 -- 277.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....formula (38) can in fact be extended to a full asymptotic expansion of ff n ; cf. 30, Theorem 2] The latter result can be viewed as an analogue of Stirling s expansion (36) and the identity (37) for polynomials of higher degree. Its proof, which is based on the work of Harris and Schoenfeld [19], is several orders of magnitude harder than that of Theorem D, and the result itself is too technical to be stated here. However, as a consequence of these developments, a complete asymptotic expansion of the function jHom(G; H o S n )j is obtained for arbitrary G and H of the form right hand ....

B. Harris and L. Schoenfeld, Asymptotic expansion for the coefficients of analytic functions, Illinois J. Math. 12 (1968), 264 -- 277.

....close to singularities and other saddle points of the function. This happens for instance with OE(z) z e z 2 . When d is fixed, this problem is solved for large classes of functions (called admissible ) either by requiring the functions to satisfy more or less stringent conditions (see [6, 7, 9]) or by taking into account the other contributions to the integral [12] The effect of d tending to infinity when OE = f d is i) to increase the difference between the largest value of f on the contour and the other ones, ii) to make OE grow fast enough; both properties make it easier to ....

B. Harris and L. Schoenfeld. Asymptotic expansions for the coefficients of analytic functions. Illinois Journal of Mathematics, 12:264--277, 1968.

....with the problem of obtaining asymptotic information on the coefficients ff n of a function f(z) P 1 n=0 ff n z n analytic in some neighborhood of the origin. Here we mention only the paper [6] of Hayman cited in the previous paragraph and the work of Harris and Schoenfeld; cf. 7] and [8]. Hayman derives an asymptotic formula for ff n under relatively mild conditions on f(z) In their paper [8] Harris and Schoenfeld study analytic functions satisfying considerably more stringent regularity conditions and obtain for the coefficients of these functions complete asymptotic ....

....n=0 ff n z n analytic in some neighborhood of the origin. Here we mention only the paper [6] of Hayman cited in the previous paragraph and the work of Harris and Schoenfeld; cf. 7] and [8] Hayman derives an asymptotic formula for ff n under relatively mild conditions on f(z) In their paper [8] Harris and Schoenfeld study analytic functions satisfying considerably more stringent regularity conditions and obtain for the coefficients of these functions complete asymptotic expansions. For the convenience of the reader we briefly explain here a special case of their result, which will be ....

B. Harris and L. Schoenfeld, "Asymptotic expansion for the coefficients of analytic functions," Illinois J. Math. 12 (1968), 264-277.

....then f n n s Gamma1 Gamma(s) L(n) Note that if F (z) has a finite dominant singularity not at z = 1, the above theorem can be applied normalizing it. For entire functions and those having nonpolar singularities at which they become very large, saddle point methods are well suited [Hay56, HS68, OR85, Hof87] The key idea is to determine a saddle point (depending on n) which is a point of minimum modulus such that the derivative of F (z) z n 1 vanishes at it. Then it has to be proven that Cauchy s integral over a contour crossing the saddle point can be splitted in two parts : an ....

B. Harris and L. Schoenfeld. Asymptotic expansions for the coefficients of analytic functions. Illinois J. Math., 12:264--277, 1968.

....the asymptotic formula (29) can be extended to a full asymptotic expansion of ff n ; cf. 41, Theorem 2] The latter result can be seen as an analogue of Stirling s expansion (27) and the identity (28) for polynomials of higher degree. Its proof, which is based on the work of Harris and Schoenfeld [21], is several orders of magnitude harder than that of Theorem 4, and the result itself is too technical to be stated here. However, as a consequence of these developments, a complete asymptotic expansion of the function jHom(G; H o S n )j is obtained for arbitrary finite groups G and H in terms of ....

B. Harris and L. Schoenfeld, Asymptotic expansion for the coefficients of analytic functions, Illinois J. Math. 12 (1968), 264--277.

....of algebraic singularities on its circle of convergence, the method of Darboux can often be successfully applied [1, Section 6] When the circle of convergence is a natural boundary, or when F(x) is an entire function, these techniques do not apply. However, the methods of Harris and Schoenfeld [8] and Hayman [10] can frequently be applied. Hayman [10] obtained rough asymptotic expansions for the coefficients of a wide class of analytic functions satisfying fairly mild regularity conditions, which he called admissible functions, and which we will refer to as H admissible functions. Harris ....

....can frequently be applied. Hayman [10] obtained rough asymptotic expansions for the coefficients of a wide class of analytic functions satisfying fairly mild regularity conditions, which he called admissible functions, and which we will refer to as H admissible functions. Harris and Schoenfeld [8] studied analytic functions satisfying considerably more stringent regularity conditions (we will refer to them as HS admissible functions) and obtained for the coefficients of these functions complete asymptotic expansions. Harris Schoenfeld type expansions do not hold for all H admissible ....

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B. Harris and L. Schoenfeld, Asymptotic expansions for the coefficients of analytic functions, Illinois J. Math., 12(1968), 264-277.

....exponential growth (2.11) of the y h (z) which follows from that nonnegativity. Further, one does not have to restrict attention to polynomials P(z , y) in the recurrence (2. 5) However, when P(z , y) is not a polynomial, one can often obtain asymptotic estimates by using more classical estimates [18,19,29]. 3. Average heights of trees In the preceding section we presented some results on the enumeration of trees of a given height, so that we held the height fixed and varied the size of the trees. In many cases what is needed, though, is information about the distribution of heights among trees of a ....

B. Harris and L. Schoenfeld, Asymptotic expansions for the coefficients of analytic functions, Illinois J. Math., 12 (1968), 264-277.

....automatically H admissibility from Theorem 9, then insert the saddle point formula of Theorem 8. An unfortunate drawback of Hayman s method is that it provides only the main term of the expansion of coefficients. Another class of functions was subsequently introduced by Harris and Schoenfeld [44]. By imposing more stringent conditions on the functions in their class, called the class of HS admissible functions, they were able to derive a full asymptotic expansion of function coefficients. As such, the HS admissible functions did not lend themselves to a direct implementation until ....

Harris, B., and Schoenfeld, L. Asymptotic expansions for the coefficients of analytic functions. Illinois Journal of Mathematics 12 (1968), 264--277.

.... n (1 Gamma1=p)n exp i Gamma p Gamma 1 p n n 1=p j (n 1) where K p = p Gamma1=2 for primes p 2 and K 2 = 2 Gamma1=2 e Gamma1=4 : More recently, Wilf [Wi] has found a (somewhat implicit) asymptotic formula for cyclic G: In [M5] building on the work of Harris and Schoenfeld [HS], a complete asymptotic expansion for the coefficients of entire functions of the form e P (z) is established, which is explicit in n and the polynomial P (z) As a special case we obtain an asymptotic expansion for the number jHom (G; S n )j of actions of an arbitrary finite group G on an ....

B. Harris and L. Schoenfeld, Asymptotic expansion for the coefficients of analytic functions, Illinois J. Math. 12 (1968), 264-277.

....hypotheses of our theorems can be weakened. It is not essential, for example, that all the coefficients of P(z , y) or of the y h (z) be nonnegative. What is really crucial is that the y h (z) should grow very rapidly as h on the positive real axis and should be relatively small elsewhere. cf. [6,7,14]. However, the appropriate growth conditions are not always easy to check, and so we have chosen to restrict our presentation to PNI sequences, which are easy to characterize, and which are of greatest interest in computer science and combinatorics. Condition (A) is not necessary for the success ....

....satisfying similar recurrences. It is also possible to use the methods of this paper to study recurrences such as (1.2) where the y h (z) are entire functions with nonnegative coefficients and where P(z , y) might also not be a polynomial. However, in many cases it is simpler to use the results of [6,7,14]. Finally, we mention that it should be possible to use our methods to study multivariate polynomials satisfying nonlinear recurrences. Such polynomials occur, for example, in studies of 2,3 trees [15] where one is interested in the coefficients of the polynomials A h (x , y) defined by A 0 (x , ....

B. Harris and L. Schoenfeld, Asymptotic expansions for the coefficients of analytic functions, Illinois J. Math. 12 (1968), 264-277.

....n p 2 h 00 (Rn ) 2:3) where h = log(f) Gamma (n 1) log z. Sufficient conditions for this method to be valid were given by Hayman (1956) and they can be checked by a computer. In some cases, a full expansion is available (Wyman 1959) and effective criteria for deciding this are available (Harris Schoenfeld 1968; Odlyzko Richmond 1985) These were also implemented in equivalent. Example. The number of increasing subsequences in permutations was considered by Lifschitz Pittel (1981) For instance, the permutation 524361 has 15 increasing subsequences, namely the empty sequence, each single element ....

Harris, B. and L. Schoenfeld (1968). Asymptotic expansions for the coefficients of analytic functions.

....generally obtain expansions in descending powers n or log n, but rather in functions of the implicitly defined saddle point. This situation was already encountered in the dominant term analysis of Bell numbers. Sets of conditions leading to full expansions have been given by Harris and Schoenfeld [14]. Odlyzko and Richmond showed in [24] that for any Hayman admissible function, a full expansion of e f(z) can be determined by Harris and Schoenfeld s method. The later situation covers for instance the 38 CHAPTER 6. SADDLE POINT ASYMPTOTICS Bell numbers. In combinatorial practice, full ....

....exposed here (see also [30] is notable in its generality as it considered saddle point analysis from a more abstract perspective by introducing general closure theorems. A similar thread was followed by Harris and Schoenfeld who gave stronger conditions then permitting full asymptotic expansions [14]; Odlyzko and Richmond [24] 50 CHAPTER 6. SADDLE POINT ASYMPTOTICS were successful in connecting this with Hayman admissibility. Another valuable work is Wyman s extension to nonpositive functions [31] Interestingly enough, developments that parallel the ones in combinatorial analysis have ....

Harris, B., and Schoenfeld, L. Asymptotic expansions for the coefficients of analytic functions. Illinois Journal of Mathematics 12 (1968), 264--277.

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B. Harris and L. Schoenfeld [1968]. "Asymptotic Expansions for the Coefficients of Analytic Functions", Illinois J. Math. 12, 1968, 264-277.

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