Philippe Flajolet, Xavier Gourdon, and Philippe Dumas. Mellin transforms and asymptotics: Harmonic sums. Theoretical Computer Science, 144(1--2):3--58, June 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....that, an adaptation of Knuth s bootstrapping method allows to approximate the average value of the number of iterations of von Neumann s algorithm by a series involving exponential functions. The asymptotic study of such series is, however, well known (by Mellin transform techniques, see e.g. [2]) leading to the results. 2. q, d) Expansions 2 # 1. It is well known that every integer x has a unique (q, d) expansion , x j 1 , where x j 0 only holds for finitely many j. For two integers x and y with (q, d) expansions x and y we define (z, c) add(x, y) by c 0 ....

....# 2 a n n ) and ) respectively. For k k 3 , we have p nk t nk p nk 3 t nk ) Noting that (1 t nk ) is exponentially small for k n and adding up the errors, we obtain p nk ) exp( n# a . We note that p n0 = O(n 2 ) It is well known (see e.g. [2]) that e x a = log a x 1 2 #(log a x) O with the periodic function #(x) given in (14) Setting x = n#, we get (13) # To apply this lemma, we note that s 0 (1) d) 0, s 1 (1) 0, # = 1, and that the roots of s 0 are q (1 d) q and q (q d) q. ....

Ph. Flajolet, X. Gourdon, and Ph. Dumas, Mellin transforms and asymptotics: Harmonic sums, Theoret. Comput. Sci. 144 (1995), 3--58.

.... ln m (or l ln(m= ln(m) we have (1 y=m) e (1 O(y =m) which yields l ln(m= ln(m) l ln(m= ln(m) m2 11 The last sum above is an ideal candidate for the Mellin transform. The main properties of the Mellin transform are discussed in Appendix A (cf. also [4, 11]) Let We are interested in the asymptotic expansion of S(x) as x = m 1, where x is a positive real number. Observe that under property (M3) from Appendix A the function S(x) falls under the harmonic sum paradigm and then the Mellin transform S (s) of S(x) is (s) 1 (s) 0; ....

.... (m 1) 2 2 l n 1 = t n;m 1 O(m2 log m) This proves Theorem 4. 17 APPENDIX: Properties of the Mellin Transform For the readers convenience, we collected here the main properties of the Mellin transform. For details and proofs see [4, 11]. M1) Direct and Inverse Mellin Transforms. Let c belong to the fundamental strip de ned below. s) M(f(x) s) Z 1 f(x)x s 1 dx ( f(x) 1 Z c i1 c i1 (A.1) M2) Fundamental Strip. The Mellin transform of f(x) exists in the fundamental strip (s) 2 ( where f(x) O(x ....

P. Flajolet, X. Gourdon, and P. Dumas, Mellin Transforms and Asymptotics: Harmonic sums, Theoretical Computer Science, 144, 3-58, 1995.

....most interesting case, and the hardest. It turns out that this case can be handled by a special analytic tool, namely the Mellin transform. The Mellin transform found myriad of applications in the analysis of algorithms. The reader is referred to an excellent survey by Flajolet, Gourdon and Dumas [9] (cf. 10, 15] For reader convenience, we collected the most important properties of the Mellin transform in Section A.4. In particular, the de nition of Mellin transform is given in (34) Property (M2) de nes the so called fundamental strip of the complex plane where the Mellin transform ....

....(F (k) F (k 1) F (k) aN N Z 1 dx O = aN N 1 O The lower bound is derived in a similar fashion. A.4 Main Properties of Mellin Transform For the reader convenience, we collected here the main properties of the Mellin transform. For details and proofs see [9, 15]. M1) Direct and Inverse Mellin Transforms. Let c belong to the fundamental strip de ned below. Then (s) M(f(x) s) f(x)x s 1 dx ( f(x) 1 Z c i1 c i1 (s)x ds: 34) M2) Fundamental Strip. The Mellin transform of f(x) exists in the fundamental strip (s) 2 ( ....

P. Flajolet, X. Gourdon, and P. Dumas, \Mellin Transforms and Asymptotics: Harmonic sums," Theoretical Computer Science, 144, 3-58, 1995.

....by the National Research Foundation under grant number 2053740. strictly greatest part in the rst position, 2) The term z from k = 1 above represents the unique composition of 1. To estimate the coecient fn of z in f(z) we follow the approach of Flajolet, Gourdon and Dumas [1], who considered a similar generating function relating to the longest run of 1 s in a random binary string of length n. fn where (x) is a continuous periodic function of period 1, mean zero, small amplitude and Fourier expansion (x) k 6=0 2k ix Proof. Let k be the smallest ....

P. Flajolet X. Gourdon and P. Dumas. Mellin transforms and asymptotics: Harmonic sums. Theoretical Computer Science, 144:3-58, 1995.

....of C 2 as follows: assume # = 2 . We derive 1 j(j 1) 1 1 2 3 2.# (log (41) C 3 : 0.419422441795 . A still more precise analysis can be done. Indeed, L(#) is a dyadic sum, which, for large #, can be analyzed using Mellin transforms: see Flajolet et al. [7]. It is well known that the dominant value is given by some function of log(#) The oscillatory part has a very small amplitude, usually of order 10 5 . Indeed, set f(y) ln(1 y) We obtain L(#) f(1 # 2 the Mellin transform of which is L # (s) s sin(#s) 1 . defined in the ....

P. Flajolet, X. Gourdon, and P. Dumas. Mellin transforms and asymptotics: Harmonic sums. Theoretical Computer Science, 144:3--58, 1995.

....The local limit theorems are obtained by refining the asymptotic estimates and the saddle point method. Due to the presence of periodic fluctuations, the analysis required for the asymptotics of E(u is more delicate. The major analytic tools used are: Euler transform, Mellin transform (cf. [7]) and the singularity analysis of Flajolet and Odlyzko [10] Such an analysis is reminiscent of that in Flajolet and Richmond [11] and Flajolet et al. 9] for quadtrees and digital search trees. As in these problems, the introduction of Euler transform is the Eureka to the problem and largely ....

.... behavior of f(w; u) as w 1 and j arg wj , which in turn will characterize the behavior of F (z; u) as z 1 in the cut plane C n [1; 1) After the introduction of Euler transform, there are typically two different approaches of obtaining the required asymptotic estimate: Mellin transform (cf. [7]) and Rice s difference formula (cf. 9, 12] We will use the first one which is more general. Write f(w; u) e OE(w;u) OE(w; u) Using the Mellin inversion formula (by an integration by parts and by beta type integral) u Gamma 1)px 1 x x Gammas Delta ds ....

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P. Flajolet, X. Gourdon, and P. Dumas, Mellin transforms and asymptotics: harmonic sums, Theoretical Computer Science, 144 (1995), 3--58.

....then from Meinardus s original theorem. For techniques leading to the required (better) error terms, see Appendix. Let D 1 (s) # 1 ) s and D 2 (s) # 2 1 ) s # 2 ) s for s #. Analytic continuations of these two functions in terms of D can be obtained in the usual way (cf. [6]) We have D 1 (s) D(s) D 2 (s) # 2 # 1 ) s D(s) the right hand side providing the required analytic continuation for 0 . Note that when 0 # 1 both functions so continued have a simple pole at s = # 1 (if # 1 0 ) and that when # = 1 the ....

P. Flajolet, X. Gourdon, and P. Dumas, Mellin transforms and asymptotics: harmonic sums, Theoretical Computer Science, 144 (1995), 3--58.

....r n;k k k (1 k ) k2 . Consequently, ff Gamma 1 Gamma (1 k ) Gamman Delta Sigma 1 Sigma 2 ; where r n;k Gammax x 1 dx n 7 R k 2 n In a similar way, we have It remains to evaluate the harmonic sum (cf. [17]) S : Consider first the case ff Gamma1. Taking K = Gamma 2 log 2 log log we obtain easily S Gamma i( Gammaff) Gamma 1kK k K A k e K 1 K log log 2 (log n) thus proving (6) We are left with the case ff Gamma1 for which we apply ....

....the case ff Gamma1. Taking K = Gamma 2 log 2 log log we obtain easily S Gamma i( Gammaff) Gamma 1kK k K A k e K 1 K log log 2 (log n) thus proving (6) We are left with the case ff Gamma1 for which we apply the Mellin inversion formula (cf. [17]) Gammaw Z Gamma 2 Gammai1 Gamma(s)w ds ( w 0) giving S = 1 Gammai1 U ff (s)ds; 11) Gamma(k 1)s The function U ff is essentially a special case of the Lerch zeta function (cf. 13] From known properties of the Lerch zeta function, it follows that, for fixed ....

[Article contains additional citation context not shown here]

P. Flajolet, X. Gourdon, and P. Dumas. Mellin transforms and asymptotics: harmonic sums. Theoretical Computer Science 144 (1995) 3--58.

....terms in the asymptotic expansion are of order (log z) j=2 in lieu of (log z) To avoid technical complications, we contend ourselves with the proof of Theorems 1 3. The in nite products of e G and of e F (when taking logarithm) are special classes of the so called harmonic sums, see [4] for a general introduction and survey. 2 The proof of the Theorems Proof of Theorem 1. Let (x) p x 1 denote the number of primes x for x 2 and (x) 0 for x 2. By writing f (de ned in (5) as a Stieltjes integral and by an integration by parts, we have f(z) z(z 1) x(x 1) x ....

....log( d (log jzj) Finally, 9) follows from the expansion (cf. 2, Eq. 5) p. 51] sin z j B 2j (2j) 1 2 2j 1 )z 2j ; jzj ) This completes the proof. Proof of Theorem 3. Sketch) By de nition, I k the right hand side is a harmonic sum (cf. [4]) and its Mellin transform (cf. 20, p. 193] is given by M [ f ; s] s sin s ) 2 s 1) Thus 1 )ds: Using the expansion (cf. 11) log 1 1 s K q O (j1 sj) s 1) and proceeding along the same lines of argument as above (with much simpler analysis) we deduce ....

P. Flajolet, X. Gourdon, and P. Dumas, Mellin transforms and asymptotics: harmonic sums, Theoretical Computer Science, 144 (1995), 3-58.

....with Gamma y and r 0. These estimates will be slightly more general than our need for the proof of Theorem 1 since some of them will be required when establishing the corresponding local limit theorem. Let f(u; log Q(u; e The sum on the right hand side being a harmonic sum (cf. [10]) we have available the Mellin inversion formula: f(u; 1 D(s)Y (u; s) ds; 8) for e 0, where Y (u; s) is the Mellin transform of the function log(1 ue ) cf. 3) Note that, for juj 1 and e s 0, Y (u; s) satisfies Y (u; s) Gamma(s) s 1 ; 9) a representation no ....

P. Flajolet, X. Gourdon, and P. Dumas, Mellin transforms and asymptotics: harmonic sums, Theoretical Computer Science, 144 (1995), 3--58.

....follow then from Meinardus s original theorem. For techniques leading to the required (better) error terms, see Appendix. Let D 1 (s) and D 2 (s) 2 Gamma 1 ) for e s ff. Analytic continuations of these two functions in terms of D can be obtained in the usual way (cf. [6]) We have D 1 (s) D(s) Gamma D 2 (s) 2 Gamma 1 ) D(s) Gamma 1 Gamma the right hand side providing the required analytic continuation for e s Gammaff 0 . Note that when 0 ff 1 both functions so continued have a simple pole at s = ff Gamma 1 (if ff ....

P. Flajolet, X. Gourdon, and P. Dumas, Mellin transforms and asymptotics: harmonic sums, Theoretical Computer Science, 144 (1995), 3--58.

....# and r 0. These estimates are slightly more general than our need for the proof of Theorem 1 since some of them will be required when establishing the corresponding local limit theorem. Let f(u, #) log Q(u, e # ) 1 ue k# . The sum on the right hand side being a harmonic sum (see [10]) we have available the Mellin inversion formula: f(u, #) 1 D(s)Y (u, s)# s ds, 9) # 0, where Y (u, s) is the Mellin transform of the function log(1 ue x ) see (4) Note that, 1 and s 0, Y (u, s) satisfies Y (u, s) #(s) s 1 , 10) a representation no longer useful ....

P. Flajolet, X. Gourdon, and P. Dumas, Mellin transforms and asymptotics: harmonic sums, Theoretical Computer Science, 144 (1995), 3--58.

....distribution is highly concentrated around OE n OE n=2 c(log n) OE n=4 c c(log n) n Z log 2 n dx; from which the estimate (7) follows. 7 Case (i) n = c(log n) In this case, Psi(z) n . Since (6) is a harmonic sum (see Flajolet et al. [10]) we have the integral representation 1 Z Gamma1=2 i1 Gamma1=2 Gammai1 Gammas Psi (s) s ds; 8) where (s) Psi(y)y We need the asymptotic behavior of e Psi(y) By Stirling s formula, we have the local limit theorem for Poisson distribution ....

.... c(log z) log jzj) c log log z (11) uniformly as jzj 1 in the sector j arg zj =3. Technically, we move the line of integration to a suitable Hankel type contour around the origin, and then evaluate the integral by an argument similar to the singularity analysis; see [10, 11]. We need a uniform estimate for =3 j arg zj . By (6) we have, writing z = re it j Phi(z) Gamma Psi(z)j )r cos t )r=2 e Gammar=4 ( Phi(r) Gamma Psi(r) 12) uniformly for =3 jtj . By Cauchy s integral formula, we have I jzj=n ( Phi(z) Gamma Psi(z) ....

P. Flajolet, X. Gourdon and P. Dumas, Mellin transforms and asymptotics: harmonic sums, Theoretical Computer Science, 144 (1995), 3--58.

....taking a circle of radius n and uses Stirling s formula for the approximation of the quantity n =n which occurs. It is suggestive to use a new name R(z) for P and consider it to be an auxiliary (and known) function; R(z) 3.2) We compute the Mellin transform of (3. 2) see [2] for de nitions and properties) s) R The function m(z) can be recovered from this by Mellin s inversion formula, 1 where 0 c log q d. By shifting the integral to the right and taking the negative residues into account, we get the desired asymptotic behaviour ....

....is again the technique of depoissonization. We set b(z) and (z) b(zd ) 1 We then get ba(z) c(q z) b(q z) As ba(0) 0, we nally obtain c(d z) d z) 4.1) We compute the Mellin transform of (4. 1) since it is a harmonic sum (see [2] for more background) we obtain ba(z) z) ns 1 d s (s) The Mellin transform s 1 can be expressed by Hurwitz zeta functions: Recall that the Hurwitz zeta function is de ned as (s; a) k a) for (s) 1 and a 0. The classical formula ....

P. Flajolet, X. Gourdon, and P. Dumas. Mellin transforms and asymptotics: Harmonic sums. Theoret. Comput. Sci., 144:3-58, 1995.

....of applications to other problems of similar nature. Besides divide and conquer algorithms, we refer the reader to [16] and the references therein for parameters in tries and its variants. A rather complete survey on Mellin transforms with an emphasis on algorithmic analysis can be found in [4]. There exist many other variants of mergesort, for example, the queue mergesort (cf. 5] and the bottom up mergesort (cf. 14] The number of comparisons used by these variants and the top down mergesort considered here all satisfy the following pattern of divide and conquer recurrence: X n = ....

P. Flajolet, X. Gourdon and P. Dumas, Mellin transforms and asymptotics: harmonic sums, Theoretical Computer Science, 144 (1995) 3--58.

....distribution is highly concentrated around n 2, # n 2 c(log n) # n 4 c c(log n) n log 2 n dx, from which the estimate (7) follows. 7 Case (i) # n = c(log n) In this case, #(z) n#2 c(log n) n . Since (6) is a harmonic sum (see Flajolet et al. [10]) we have the integral representation e z #(z) 1 1 2 i# 1 2 i# z s ds, 8) where # # (s) e y #(y)y dy. We need the asymptotic behavior of e y #(y) By Stirling s formula, we have the local limit theorem for Poisson distribution e y = y 1 (1 x ....

....# # # # # # # c(log z) log z ) c log log z (11) uniformly as z # # in the sector # 3. Technically, we move the line of integration to a suitable Hankel type contour around the origin, and then evaluate the integral by an argument similar to the singularity analysis; see [10, 11]. We need a uniform estimate for # 3 # #. By (6) we have, writing z = re it #(z) #(z) 1 2 k )r cos t (1 2 k )r 2 e r 4 (#(r) #(r) 12) uniformly for # 3 # t # #. By Cauchy s integral formula, we have # z =n z n 1 (#(z) #(z) dz. Split the ....

P. Flajolet, X. Gourdon and P. Dumas, Mellin transforms and asymptotics: harmonic sums, Theoretical Computer Science, 144 (1995), 3--58.

....expressions for S n;k which completely characterize the asymptotic behaviors of S n;k for 1 6 k 6 n, as n 1. The rst expression extends the domain of validity of (4) to 1 6 k = o(n ) Our proof follows a similar line of generating function but with an appeal to Mellin transforms (cf. [9]) This approach is computationally simpler. We then propose another uniform asymptotic expression for S n;k for k 1 and k 6 n, as n 1 using an elementary argument. It should be noted that uniform asymptotic expressions are especially useful for practical purposes since in reality it is not ....

....d (c 0) j j ( 62 (1; 0] Analytic continuation of B k (z) follows from that of B(z) which in turn is obtained from that of T (z) see Flajolet and Odlyzko [12] or Corless et al. 5] We now make explicit the local behavior of ) as 0. From the Mellin inversion formula (cf. [9]) 1 (s)w ds ( w 0; a 0) it follows, by absolute convergence, that 1 (s) Y k (s) ds (a k ) 14) Y k (s) s k 2 ) The singularities of Y k (s) will be determined by the asymptotic behavior of j = j k) as j 1. By Stirling s formula k s ....

P. Flajolet, X. Gourdon, and P. Dumas, Mellin transforms and asymptotics: harmonic sums, Theoretical Computer Science, 144 (1995), 3-58.

....approximate expressions for S n,k which completely characterize the asymptotic behaviors of S n,k for 1 # k # n, as n # #. The first expression extends the domain of validity of (4) to 1 ) Our proof follows a similar line of generating functions but with an appeal to Mellin transforms (cf. [9]) This approach is computationally simpler. We then propose another uniform asymptotic expression for S n,k for k k # n, as n using an elementary argument. It should be noted that uniform asymptotic expressions are especially useful for practical purposes since in reality it is not obvious ....

.... 1 # )B k (e 1 # ) d# (c 0) Analytic continuation of B k (z) follows from that of B(z) which in turn is obtained from that of T (z) see Flajolet and Odlyzko [12] or Corless et al. 5] We now make explicit the local behavior of B k (e 1 # ) as # 0. From the Mellin inversion formula (cf. [9]) e w = 1 #(s)w s ds (#w 0, a 0) it follows, by absolute convergence, that B k (e 1 # ) 1 #(s)# s Y k (s) ds (a k ) where Y k (s) #s k ) The singularities of Y k (s) will be determined by the asymptotic behavior of e j (j k) as j ##. By ....

P. Flajolet, X. Gourdon, and P. Dumas, Mellin transforms and asymptotics: harmonic sums, Theoretical Computer Science, 144 (1995), 3--58.

....r n,k k# k (1 # k ) n k2 k e n 2 . Consequently, 1 # k ) n # 1 # 2 , where # 1 : r n,k 2 k e n 2 2 x e n 2 x 1 7 # 2 : R k 2 n . In a similar way, we have . It remains to evaluate the harmonic sum (cf. [17]) S : Consider first the case # 1. Taking K = 1) 2 log 2 log log we obtain easily S #( #) # # 1#k#K k K K 1 K log log (log n) thus proving (6) We are left with the case # # 1 for which we apply the Mellin inversion ....

..... Consider first the case # 1. Taking K = 1) 2 log 2 log log we obtain easily S #( #) # # 1#k#K k K K 1 K log log (log n) thus proving (6) We are left with the case # # 1 for which we apply the Mellin inversion formula (cf. [17]) e w = 1 #(s)w s ds (#w 0) giving S = 1 1 U # (s)ds, 11) U # (s) 2 (k 1)s (#s 0) The function U # is essentially a special case of the Lerch zeta function (cf. 13] From known properties of the Lerch zeta function, it follows that, for fixed # R, the ....

[Article contains additional citation context not shown here]

P. Flajolet, X. Gourdon, and P. Dumas. Mellin transforms and asymptotics: harmonic sums. Theoretical Computer Science 144 (1995) 3--58.

....The local limit theorems are obtained by refining the asymptotic estimates and the saddle point method. Due to the presence of periodic fluctuations, the analysis required for the asymptotics of E(u is more delicate. The major analytic tools used are: Euler transform, Mellin transform (cf. [7]) and the singularity analysis of Flajolet and Odlyzko [10] Such an analysis is reminiscent of that in Flajolet and Richmond [11] and Flajolet et al. 9] for quadtrees and digital search trees. As in these problems, the introduction of Euler transform is the Eureka to the problem and largely ....

.... behavior of f(w, u) as w arg w #, which in turn will characterize the behavior of F (z, u) as z 1 in the cut plane C [1, #) After the introduction of Euler transform, there are typically two di#erent approaches of obtaining the required asymptotic estimate: Mellin transform (cf. [7]) and Rice s di#erence formula (cf. 9, 12] We will use the first one which is more general. Write f(w, u) e #(w,u) Using the Mellin inversion formula (by an integration by parts and by beta type integral) 1)px 1 x # x s ds ( 1 a 0) we have ....

[Article contains additional citation context not shown here]

P. Flajolet, X. Gourdon, and P. Dumas, Mellin transforms and asymptotics: harmonic sums, Theoretical Computer Science, 144 (1995), 3--58.

....of applications to other problems of similar nature. Besides divide and conquer algorithms, we refer the reader to [16] and the references therein for parameters in tries and its variants. A rather complete survey on Mellin transforms with an emphasis on algorithmic analysis can be found in [4]. There exist many other variants of mergesort, for example, the queue mergesort (cf. 5] and the bottom up mergesort (cf. 14] The number of comparisons used by these variants and the top down mergesort considered here all satisfy the following pattern of divide and conquer recurrence: X n = ....

P. Flajolet, X. Gourdon and P. Dumas, Mellin transforms and asymptotics: harmonic sums, Theoretical Computer Science, 144 (1995) 3-58.

....terms in the asymptotic expansion are of order (log z) j 2 in lieu of (log z) j . To avoid technical complications, we contend ourselves with the proof of Theorems 1 3. The infinite products of G and of F (when taking logarithm) are special classes of the so called harmonic sums, see [4] for a general introduction and survey. 2 The proof of the Theorems Proof of Theorem 1. Let #(x) p#x 1 denote the number of primes x for x and #(x) 0 for x 2. By writing f (defined in (5) as a Stieltjes integral and by an integration by parts, we have 1) 2 x(x 1) x z) ....

....# #log(# # ) # d# # # . Finally, 9) follows from the expansion (cf. 2, Eq. 5) p. 51] sin z B 2j (2j) 2j 1 )z 2j , z #) This completes the proof. Proof of Theorem 3. Sketch) By definition, k#1 I k the right hand side is a harmonic sum (cf. [4]) and its Mellin transform (cf. 20, p. 193] is given by M [ f ; s] s sin #s ) 2 1) Thus 1 #z s )ds. Using the expansion (cf. 11) log 1 1 s K q O ( 1 s ) s # 1) and proceeding along the same lines of argument as above (with much simpler ....

P. Flajolet, X. Gourdon, and P. Dumas, Mellin transforms and asymptotics: harmonic sums, Theoretical Computer Science, 144 (1995), 3--58.

....For the range h n we use the Taylor expansion again to obtain fi fi = O(nq 2h ) 5.8) Inserting (5.6) 5.7) and (5.8) into (5.5) we obtain EM n = 1 Gamma exp( Gammanpq O(1)O e Gamman log n O(n log n) O(n (5. 9) We now apply the Mellin transform (cf. [2, 4]) to the function f(t) h1 (1 Gamma exp( Gammatpq ) This yields the transformed function f (s) Gamma Gamma(s)p ; for Gamma 1 s 0: 5.10) Application of the Mellin inversion formula, shifting the line of integration to the right and collecting residues yields f(t) ....

P. Flajolet, X. Gourdon, and P. Dumas, Mellin transforms and asymptotics: Harmonic sums, Theoretical Computer Science 144 (1995), 3--58.

....102 ; n 13; They are all easy to prove by considering 1 1 Gamma oe(zv; v Gamma1 and looking for the coefficient of a fixed power of v. 4. The largest part in Carlitz compositions For ordinary partitions, the statistic largest part of a composition has obtained a lot of attention, [5,10]. Now we want to sketch the analogous analysis for the case of Carlitz compositions. Let us first consider the generating function(s) where all parts are less than or equal to h. Then essentially the same idea as in ( works, except that we only use a factor (zu) Delta Delta Delta ....

....that the largest part is h we have then approximately and to get the desired average value En we must sum this up over h 0. The next step is to use the exponential approximation (1 Gamma a) e Gammaan En h0 1 Gamma e Gammanae =oe But this quantity is quite well studied [5] (we might even set N : n=oe (ae) for the moment to make it look closer to already existing formulae) The answer is En log 1=ae N Gamma 1 2 ffi(log 1=ae N) with a certain periodic function ffi(x) that has period 1, mean 0, and small amplitude. Rewriting this we find En log 1=ae n ....

P. Flajolet, X. Gourdon, and P. Dumas, Mellin transform and asymptotics: Harmonic sums, Theoretical Computer Science 144 (1995), 3--58.

....(4.18) with G(0) 1. Taking the logarithm of (4.18) with F (z) log G(z) we obtain F (2z) Gamma F (z) log ; 4.19) with F (0) 0. Functional equations of the type (4. 19) are often encountered in the analysis for algorithms and they are usually handled by the Mellin transform [11]. The interested reader can find more on Mellin transform in a recent survey [11] The Mellin transform F (s) of a real valued function F (z) is defined as F (z)z dz = M[F ; s] and its inverse is 1 Z c i1 c Gammai1 (s)ds = M [F ; z] where c belongs to the so called ....

....we obtain F (2z) Gamma F (z) log ; 4.19) with F (0) 0. Functional equations of the type (4. 19) are often encountered in the analysis for algorithms and they are usually handled by the Mellin transform [11] The interested reader can find more on Mellin transform in a recent survey [11]. The Mellin transform F (s) of a real valued function F (z) is defined as F (z)z dz = M[F ; s] and its inverse is 1 Z c i1 c Gammai1 (s)ds = M [F ; z] where c belongs to the so called fundamental strip where the Mellin transform is analytic (cf. 11] Taking the Mellin ....

[Article contains additional citation context not shown here]

P. Flajolet, X. Gourdon, P. Dumas, Mellin Transforms and Asymptotics: Harmonic Sums, Theoretical Computer Science, 144, 3-58, 1995.

....most interesting case, and the hardest. It turns out that this case can be handled by a special analytic took, namely the Mellin transform. The Mellin transform found myriad of applications in the analysis of algorithms. The reader is referred to an excellent survey by Flajolet, Gourdon and Dumas [7] (cf. 8, 10] For reader convenience, we collected the most important properties of the Mellin transform in Section A.3. In particular, the denition of Mellin transform is given in (33) Property (M2) denes the so called fundamental strip of the complex plane where the Mellin transform exists. ....

....obtain Gamma a ( Gamma 2ik= ln V ) Gamma( Gamma (2ik) ln V ) Combining everything we nally prove Corollary 4. A.3 Main Properties of Mellin Transform For the reader convenience, we collected here the main properties of the Mellin transform. For details and proofs see [7, 10]. M1) Direct and Inverse Mellin Transforms. Let c belong to the fundamental strip dened below. s) M(f(x) s) f(x)x s Gamma1 dx ( f(x) Z c i1 c Gammai1 (s)x (M2) Fundamental Strip. The Mellin transform of f(x) exists in the fundamental strip (s) 2 ( Gammaff; Gammafi) ....

P. Flajolet, X. Gourdon, and P. Dumas, Mellin Transforms and Asymptotics: Harmonic sums, Theoretical Computer Science, 144, 358, 1995.

....(v 2 (n) 1)u n : 6) Here v 2 (n) denotes the dyadic valuation of n, i.e. the number of positive divisors of n which are a power of two. Now, everything is prepared to determine an asymptotic equivalent for the average Horton Strahler number. We use the Mellin summation method as described in [FGD95] to evaluate the sum. For that purpose we set u = exp( t) and apply the well known identity exp( tj) 1 2 i Z c i1 c i1 (s)j s t s ds; i 2 = 1; for some c in the fundamental strip of the Mellin transform of exp( tj) and for (s) the complete gamma function. This is how it is possible to ....

P. Flajolet, X. Gourdon and P. Dumas, Mellin transforms and asymptotics: Harmonic sums, Theoretical Computer Science 144, 358, 1995

.... consider the series ( appearing in (6) The analysis of harmonic summations like ( is performed by means of the Mellin transform which is by now a fairly well understood methodology in analytic combinatorics and analysis of algorithms (see for instance the excellent survey by Flajolet et al. [13]) going back to the seminal paper of De Bruijn et al. 3] We determine the Mellin transform of (e t ) which proves to be given in closed form by (s) s) s) 1 1 2 s ; s) 1; where (s) is the complete gamma function and (s) is the Riemann zeta function. Then, according to the ....

P. Flajolet, X. Gourdon and P. Dumas, Mellin transforms and asymptotics: Harmonic sums, Theoretical Computer Science 144, 3-58, 1995

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Philippe Flajolet, Xavier Gourdon, and Philippe Dumas. Mellin transforms and asymptotics: Harmonic sums. Theoretical Computer Science, 144(1--2):3--58, June 1995.

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Flajolet, P., Gourdon, X., and Dumas, P. Mellin transforms and asymptotics : Harmonic sums. Theoretical Computer Science 144, 12 (June 1995), 358.

....Mapping. Assume that f (s) O(jsj ) with r 1, as jsj 1 and that f (s) admits the singular expansion (28) for (s) 2 ( M; Then f(x) satis es the asymptotic expansion (28) at x = 0 . Fig. 1. Main properties of the Mellin transform. falls under the harmonic sum paradigm of [8]. The Mellin approach is by now a standard technique in the analysis of algorithms. For the reader s convenience, we recall its main properties in Fig. 1, following [8] to which we refer globally for detailed validity conditions. First, the Mellin transform L (s) M(L(e ) s) of ) is ....

....expansion (28) at x = 0 . Fig. 1. Main properties of the Mellin transform. falls under the harmonic sum paradigm of [8] The Mellin approach is by now a standard technique in the analysis of algorithms. For the reader s convenience, we recall its main properties in Fig. 1, following [8], to which we refer globally for detailed validity conditions. First, the Mellin transform L (s) M(L(e ) s) of ) is computed by the harmonic sum property (M3) For (s) 2 (1; 1) the transform evaluates to (s) s) s) where (s) n 1 n is the Riemann zeta function, and ....

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P. Flajolet, X. Gourdon, P. Dumas, Mellin Transforms and Asymptotics: Harmonic Sums, Theoretical Computer Science, 144, 3-58, 1995.

.... the original function f and singularities of the transforms f ; ii) harmonic sums de ned as sums of the form f( x) have a transform that factorizes as ( f (s) The conjunction of both properties then renders possible the analysis of fairly intricate combinatorial sums: see [6] for an extensive survey and Szpankowski s book [14] for many applications to the analysis of algorithms. Property (i) results from the Mellin inversion formula and the residue theorem; Property (ii) re ects the action of Mellin transforms on rescaled functions. Proof. One can rewrite the two ....

Flajolet, P., Gourdon, X., and Dumas, P. Mellin transforms and asymptotics : Harmonic sums. Theoretical Computer Science 144, 1-2 (1995), 3-58.

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Philippe Flajolet, Xavier Gourdon and Philippe Dumas, Mellin transforms and asymptotics: harmonic sums, 55 pp., see, e.g.,the website of Flajolet.

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P. Flajolet, X. Gourdon, P. Dumas, Mellin transform and asymptotics: Harmonic sums, in Theoretical Computer Science, Vol 144, No 1-2, pp. 3-58, 1995.

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P. Flajolet, X. Gourdon and P. Dumas, Mellin transforms and asymptotics: Harmonic sums, Theoretical Computer Science, 144 (1995), 3--58.

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Philippe Flajolet, Xavier Gourdon and Philippe Dumas, Mellin transforms and asymptotics: harmonic sums, 55 pp., see, e.g.,the website of Flajolet.

No context found.

P. Flajolet, X. Gourdon, and P. Dumas. Mellin transforms and asymptotics: Harmonic sums. Theoretical Computer Science, 144:3-58, 1995.

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P. Flajolet, X. Gourdon and P. Dumas, Mellin transforms and asymptotics: Harmonic sums, Theoretical Computer Science, 144 (1995), 3--58.

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P. Flajolet, X. Gourdon, and P. Dumas. Mellin transforms and asymptotics: Harmonic sums. Theoretical Computer Science, 144:3-58, 1995.

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P. Flajolet, X. Gourdon, and P. Dumas. Mellin transforms and asymptotics: Harmonic sums. Theoretical Computer Science, 144:3-58, 1995.

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Philippe Flajolet, Xavier Gourdon, and Philippe Dumas, "Mellin transforms and asymptotics: Harmonic sums," Theoretical Computer Science 144 (1995), 3-- 58.

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P. Flajolet, X. Gourdon, and P. Dumas. Mellin transforms and asymptotics : Harmonic sums. Theoretical Computer Science, 144(1--2):3--58, 1995.

No context found.

Philippe Flajolet, Xavier Gourdon, and Philippe Dumas, \Mellin transforms and asymptotics: Harmonic sums," Theoretical Computer Science 144 (1995), 3-58.

No context found.

P. Flajolet, X. Gourdon, and P. Dumas. Mellin transforms and asymptotics: Harmonic sums. Theoretical Computer Science, 144:3-58, 1995.

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P. Flajolet, X. Gourdon, and P. Dumas, Mellin Transforms and Asymptotics: Harmonic sums, Theoretical Computer Science, 144, 3-58, 1995.

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P. Flajolet, X. Gourdon, and P. Dumas. Mellin transforms and asymptotics: Harmonic sums. Theoretical Computer Science, 144:3-58, 1995.

No context found.

P. Flajolet, X. Gourdon, and P. Dumas. Mellin transforms and asymptotics: Harmonic sums. Theoretical Computer Science, 144:3-58, 1995.

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P. Flajolet, X. Gourdon and P. Dumas, Mellin transforms and asymptotics: harmonic sums, Theoretical Computer Science, 144 (1995), 3--58.

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P. Flajolet, X. Gourdon and P. Dumas, Mellin transforms and asymptotics: harmonic sums, Theoretical Computer Science, 144 (1995), 3--58.

No context found.

Flajolet, P., Gourdon, X., and Dumas, P. Mellin Transforms and Asymptotics: Harmonic Sums. Theoretical Computer Science, 144, 3--58, 1995.

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