## Mellin transforms and asymptotics: Finite differences and Rice's integrals (1995)

Citations: | 82 - 8 self |

### BibTeX

@MISC{Flajolet95mellintransforms,

author = {Philippe Flajolet and Robert Sedgewick},

title = { Mellin transforms and asymptotics: Finite differences and Rice's integrals},

year = {1995}

}

### Years of Citing Articles

### OpenURL

### Abstract

High order differences of simple number sequences may be analysed asymptotically by means of integral representations, residue calculus, and contour integration. This technique, akin to Mellin transform asymptotics, is put in perspective and illustrated by means of several examples related to combinatorics and the analysis of algorithms like digital tries, digital search trees, quadtrees, and distributed leader election.

### Citations

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Citation Context ...epresentation of data. The most famous of the first generation examples comprise radix exchange sort, digital "tries" and digital search trees, for which we refer the reader to Knuth's descr=-=iption in [20]-=-: see pages 131--134 (radix exchange sort) and Exercise 5.2.2-54 page 138 (assigned to S. O. Rice), as well as Section 6.3 (tries, Patricia trees, and digital search trees). This work was partly suppo... |

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Citation Context ... by means of integration contours of Hankel type. The situation is analogous to the asymptotic analysis of coefficients of functions with nonpolar singularities (the method of singularity analysis of =-=[8]-=-), to the application of Mellin--Perron formulae to Dirichlet series with algebraic or logarithmic singularities [15], or to the analysis of Mellin transforms in the nonpolar case [5]. Rather than sta... |

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Citation Context ...\Gamma1) k [k] !; where [k] ! = / 1 \Gamma 2 d\Gamma1 3 d !/ 1 \Gamma 2 d\Gamma1 4 d ! \Delta \Delta \Delta / 1 \Gamma 2 d\Gamma1 n d ! ; [2] ! = 1: The analysis of quadtrees is introduced in Mahmoud =-=[21]-=-, and this particular example is borrowed from [7] where cost measures of quadtrees are treated systematically by means of Lindelof--Mellin integrals and generalized hypergeometric functions. In the c... |

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Citation Context ...ial bounds and integration, 000 (n) ! Z 1 n 1=3 dt (1 + t n ) n = O(e \Gamma 1 2 n 1=3 ): Proceeding in the same way with a triple decomposition of J! , one establishes J! = O(e \Gamma 1 2 p log n ): =-=(14)-=- (iii). Let J? denote the contribution given by the integral along the portion C ffi of C 2 [ C 3 [ C 4 defined by !(s)s\Gamma 1 p log n : By Stirling's formula the approximation !(s) = n s \Gamma(1 \... |

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Citation Context ... (1 + fi) r + \Delta \Delta \Delta + 1 (n \Gamma 1 + fi) r : These quantities thus extend the usual harmonic numbers. The expressions to be derived also involve a variant of the Bell polynomials (see =-=[3]-=- for the standard form). Let x 1 ; x 2 ; : : : be a collection of indeterminates. The modified Bell polynomials Lm = Lm (x 1 ; x 2 ; : : : ) are defined by exp / 1 X k=1 x k t k k ! = 1 + 1 X m=1 Lm t... |

187 |
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Citation Context ...rees, and distributed leader election. Introduction The problem of estimating asymptotically high order differences of some fixed numerical sequence ff k g, D n [f ] = n X k=0 / n k ! (\Gamma1) k f k =-=(1)-=- is delicate: the binomial coefficients get close to 2 n while, for many explicitly given sequences, the differences D n tend to be polynomially bounded in n, and thus exponentially smaller than impli... |

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Citation Context ...mma 1 n )Z n ; and its asymptotic behaviour is of the form Z n = cn \Gamma1=4 sin(2n 1=2 + `) + o(n \Gamma1=4 ); for some constants c; `. Alternative approaches are discussed in Odlyzko's survey: see =-=[24]-=- and references therein. 6. Mellin transforms and Rice integrals Rice's integrals entertain close ties with inverse Mellin transforms and their modus operandi is very similar. We only discuss here the... |

97 |
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- 1965
(Show Context)
Citation Context ...resist elementary attempts that rely on an asymptotic evaluation of individual terms of the sequence ff n g. The binomial sums of (1) are naturally basic objects of the calculus of finite differences =-=[16, 22, 23]-=-. They acquired interest in the community of researchers working in the average case analysis of algorithms and data structures after De Bruijn, Knuth, and Rice in the mid 1960's showed their central ... |

90 |
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(Show Context)
Citation Context ...resist elementary attempts that rely on an asymptotic evaluation of individual terms of the sequence ff n g. The binomial sums of (1) are naturally basic objects of the calculus of finite differences =-=[16, 22, 23]-=-. They acquired interest in the community of researchers working in the average case analysis of algorithms and data structures after De Bruijn, Knuth, and Rice in the mid 1960's showed their central ... |

88 |
twelve lectures on subjects suggested by his life and work
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(Show Context)
Citation Context ...nts of functions with nonpolar singularities (the method of singularity analysis of [8]), to the application of Mellin--Perron formulae to Dirichlet series with algebraic or logarithmic singularities =-=[15]-=-, or to the analysis of Mellin transforms in the nonpolar case [5]. Rather than stating general conditions that would be rather heavy, we content ourselves here with presenting in some detail the anal... |

84 |
Asymptotic Expansions: Their Derivation and Interpretation
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(Show Context)
Citation Context ...ntire function can also be treated by the method of Rice integrals in conjunction with the saddle point method. This reflects the corresponding situation for the analysis of inverse Mellin transforms =-=[4]-=-. For instance, the sequence of Kooman and Tijdeman (related to Laguerre polynomials) Z n = n X k=0 / n k ! (\Gamma1) k k! involves the entire function '(s) = (\Gamma(s)) \Gamma1 . This sequence is ho... |

64 |
Digital search trees revisited
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- 1986
(Show Context)
Citation Context ... [f ] for f k = k \Gamma , had provided the initial motivation for [12]. Some of the reasons why sums investigated here are of interest in the analysis of digital structures are surveyed in our paper =-=[13]-=- as well as in the combinatorial synthesis [10]. A follow-up to [12] was written by Szpankowski [27], and Prodinger [26] gives an amusing discussion of the method in comparison with standard Mellin tr... |

56 |
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(Show Context)
Citation Context ... are asymptotically in geometric progression growing roughly in proportion to N k = e k . The T n thus illustrate some of the peculiarities of the asymptotic analysis of holonomic sequences, of which =-=[33]-=- contains a general discussion.sThis last example demonstrates the fact that complex poles of '(s) induce periodic fluctuations in log n: if s 0 = oe 0 + i 0 , then n s0 = n oe 0 exp(i 0 log n): 4. Th... |

39 | Mellin transforms and asymptotics: digital sums. Theoret
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(Show Context)
Citation Context ... +1[. Then, the differences of the sequence f'(k)g admit the integral representation n X k=n0 / n k ! (\Gamma1) k '(k) = (\Gamma1) n 2i Z C '(s) n! s(s \Gamma 1) \Delta \Delta \Delta (s \Gamma n) ds; =-=(6)-=- where C is a positively oriented closed curve that lies in the domain of analyticity of '(s), encircles [n 0 ; n], and does not include any of the integers 0; 1; : : : ; n 0 \Gamma 1. Proof. This is ... |

31 |
Patricia tries again revisited
- Szpankowski
- 1990
(Show Context)
Citation Context ...trie recurrence (10): f n = a n + 2 2 n \Gamma 2 n\Gamma1 X k=1 / n k ! f k : Szpankowski et al. have extended it to the analysis of tries and Patricia trees under a biased Bernoulli model, see e.g., =-=[18, 28, 29, 30]-=- and references therein. Flajolet and Sedgewick [13], following Knuth, have developed the analysis of digital searching trees in this fashion: difference equations then get replaced by difference--dif... |

31 |
A Course in Modern Analysis, fourth ed
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- 1990
(Show Context)
Citation Context ...definition of modified Bell polynomials. The approximation follows from the estimate i n+1 (1; fi) = log n \Gamma \Gamma 0 (fi) \Gamma(fi) +O( 1 n ); and from standard estimates of the gamma function =-=[32]-=-: \Gamma(n + 1) \Gamma(n + 1 \Gamma ff) = n ff (1 +O( 1 n )): Example 1. Differences of inverse powers and harmonic numbers. Define the sums S n (m) = n X k=1 / n k ! (\Gamma1) k k m ; for m an intege... |

28 |
On the Average Number of Maxima in a Set of Vectors
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- 1989
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Citation Context ...wo complex poles s = \Sigmai of the function '(s) = (1 + s 2 ) \Gamma1 . is also related to the exponential integral (by taking exponential generating functions of both sides). More generally, Buchta =-=[2]-=- has shown that \GammaS n (m \Gamma 1) equals the expected number of maxima of n vectors in m--dimensional space, a problem of interest in computational geometry, and he has derived second order asymp... |

26 | Hypergeometrics and the cost structure of quadtrees
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- 1995
(Show Context)
Citation Context ...a1 3 d !/ 1 \Gamma 2 d\Gamma1 4 d ! \Delta \Delta \Delta / 1 \Gamma 2 d\Gamma1 n d ! ; [2] ! = 1: The analysis of quadtrees is introduced in Mahmoud [21], and this particular example is borrowed from =-=[7]-=- where cost measures of quadtrees are treated systematically by means of Lindelof--Mellin integrals and generalized hypergeometric functions. In the case of additive cost measures, the Euler transform... |

24 |
Handbuch der Laplace Transformation
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- 1950
(Show Context)
Citation Context ...rity analysis of [8]), to the application of Mellin--Perron formulae to Dirichlet series with algebraic or logarithmic singularities [15], or to the analysis of Mellin transforms in the nonpolar case =-=[5]-=-. Rather than stating general conditions that would be rather heavy, we content ourselves here with presenting in some detail the analysis of sums that generalize the S n (m) when m is no longer an in... |

24 | Generalized Digital Trees and their Difference-Differential Equations. Random Structures and Algorithms
- Flajolet, Richmond
- 1992
(Show Context)
Citation Context ...llowing Knuth, have developed the analysis of digital searching trees in this fashion: difference equations then get replaced by difference--differential equations, with further analyses appearing in =-=[11]-=-. Kirschenhofer, Prodinger, et al. in [17, 19] have treated in this way several multidimensional searching problems. Example 5. Extreme points in quadtrees. The analysis of the cost of searching point... |

23 |
A characterization of digital search trees from the successful search viewpoint
- Szpankowski
- 1991
(Show Context)
Citation Context ...trie recurrence (10): f n = a n + 2 2 n \Gamma 2 n\Gamma1 X k=1 / n k ! f k : Szpankowski et al. have extended it to the analysis of tries and Patricia trees under a biased Bernoulli model, see e.g., =-=[18, 28, 29, 30]-=- and references therein. Flajolet and Sedgewick [13], following Knuth, have developed the analysis of digital searching trees in this fashion: difference equations then get replaced by difference--dif... |

22 | How to select a loser - Prodinger - 1993 |

15 |
On the variance of the external path length in a symmetric digital trie
- Kirschenhofer, Prodinger, et al.
- 1989
(Show Context)
Citation Context ...trie recurrence (10): f n = a n + 2 2 n \Gamma 2 n\Gamma1 X k=1 / n k ! f k : Szpankowski et al. have extended it to the analysis of tries and Patricia trees under a biased Bernoulli model, see e.g., =-=[18, 28, 29, 30]-=- and references therein. Flajolet and Sedgewick [13], following Knuth, have developed the analysis of digital searching trees in this fashion: difference equations then get replaced by difference--dif... |

15 |
Some results on V -ary asymmetric tries
- Szpankowski
- 1988
(Show Context)
Citation Context |

12 | Some results on v-ary asymmetric tries - Szpankowski - 1988 |

10 | Multidimensional digit searching and some new parameters in tries
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- 1993
(Show Context)
Citation Context ... of digital searching trees in this fashion: difference equations then get replaced by difference--differential equations, with further analyses appearing in [11]. Kirschenhofer, Prodinger, et al. in =-=[17, 19]-=- have treated in this way several multidimensional searching problems. Example 5. Extreme points in quadtrees. The analysis of the cost of searching points with smallest x--coordinate in a randomly gr... |

9 |
Some uses of the Mellin integral transform in the analysis of algorithms. In Combinatorial algorithms on words (Maratea
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- 1984
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Citation Context ...t, the kernel of Rice integrals reduces for large n to a Mellin kernel, a property explored by Szpankowski in [27]. There are other connections (the Poisson-Mellin-Newton cycle) that are hinted at in =-=[9]-=- and briefly reviewed in Section 6. The authors are extremely grateful to Helmut Prodinger and Wojciech Szpankowski for their invitation to write down this new version of [12] in which we have added s... |

7 |
Hypothetical analyses: approximate counting in the style of Knuth, path length in the style of Flajolet
- Prodinger
- 1992
(Show Context)
Citation Context ...are of interest in the analysis of digital structures are surveyed in our paper [13] as well as in the combinatorial synthesis [10]. A follow-up to [12] was written by Szpankowski [27], and Prodinger =-=[26]-=- gives an amusing discussion of the method in comparison with standard Mellin transforms techniques. In what follows, we emphasize general methodology. Bibliographical indications relative to more rec... |

5 |
Multidimensional digital searching---alternative data structures
- Kirschenhofer, Prodinger
- 1994
(Show Context)
Citation Context ... of digital searching trees in this fashion: difference equations then get replaced by difference--differential equations, with further analyses appearing in [11]. Kirschenhofer, Prodinger, et al. in =-=[17, 19]-=- have treated in this way several multidimensional searching problems. Example 5. Extreme points in quadtrees. The analysis of the cost of searching points with smallest x--coordinate in a randomly gr... |

2 |
The asymptotic evaluation of some alternating sums involving binomial coefficients. Unpublished memoir
- Flajolet, Sedgewick
- 1983
(Show Context)
Citation Context ...isons necessary to sort a set of n random bit strings by the radix--exchange algorithm [20, Ex. 5.2.2-38]. The present paper is an expanded and updated version of an unpublished memoir of the authors =-=[12]-=- that was written and distributed around 1983. The subject gained renewed interest as similar and often closely related problems surfaced in diverse areas of the analysis of algorithms like: text sear... |

2 |
Vorlesungen uber Differenzenrechnung
- orlund
- 1924
(Show Context)
Citation Context ...resist elementary attempts that rely on an asymptotic evaluation of individual terms of the sequence ff n g. The binomial sums of (1) are naturally basic objects of the calculus of finite differences =-=[16, 22, 23]-=-. They acquired interest in the community of researchers working in the average case analysis of algorithms and data structures after De Bruijn, Knuth, and Rice in the mid 1960's showed their central ... |

2 | Multidimensional digital searching - alternative data structures, Random Struc - Kirschenhofer, Prodinger - 1994 |

1 |
Algebraic methods for trie statistics. Annals of Discrete Mathematics 25
- Flajolet, egnier, et al.
- 1985
(Show Context)
Citation Context ...tial motivation for [12]. Some of the reasons why sums investigated here are of interest in the analysis of digital structures are surveyed in our paper [13] as well as in the combinatorial synthesis =-=[10]-=-. A follow-up to [12] was written by Szpankowski [27], and Prodinger [26] gives an amusing discussion of the method in comparison with standard Mellin transforms techniques. In what follows, we emphas... |

1 |
How to select a looser. Discrete Mathematics 120
- Prodinger
- 1993
(Show Context)
Citation Context ...listic election of a loser. This example belongs to the orbit of the so--called "Patricia" variant of digital tries [20, p. 490--504]. Its treatment via Rice's integrals was developed by Pro=-=dinger in [25]-=-. The problem involves analysing the sum [25, Theorem 7]: V n = n\Gamma1 X k=1 / n k ! B k 2 k \Gamma 1 : The sum is in fact alternating since the nonzero Bernoulli numbers themselves alternate in sig... |

1 |
The evaluation of an alternative [sic!] sum with applications to the analysis of algorithms
- Szpankowski
- 1988
(Show Context)
Citation Context ...on function then contribute asymptotic terms in direct relation to their real part. In fact, the kernel of Rice integrals reduces for large n to a Mellin kernel, a property explored by Szpankowski in =-=[27]-=-. There are other connections (the Poisson-Mellin-Newton cycle) that are hinted at in [9] and briefly reviewed in Section 6. The authors are extremely grateful to Helmut Prodinger and Wojciech Szpanko... |

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1 | Vorlesungen iiber Diflerenzenrrchnung - Norlund - 1954 |

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