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Asymptotic Estimates of Elementary Probability Distributions - Hwang (2 citations) (Correct)
....888# 313 where r = m # 1. Note that this expansion can also be obtained from (3) but with more involved computations. 2.3 Poissonization Poissonization is a widely used technique in stochastic process, summability of divergent sequence, analysis of algorithms, etc. see, for example, [1, 6, 18, 35, 19]. The idea is roughly described as follows. Given a discrete probability distribution k#0 (or, in general, a complex sequence) consider the Poisson generating function: b(#) e # a j C) The usual Poisson heuristic reads: If the sequence k#0 is smooth enough, then a n ....
B. C. Berndt, Ramanujan's notebooks, part I, Springer-Verlag, New York, 1985.
Asymptotic Estimates of Elementary Probability Distributions - Hwang (1996) (2 citations) (Correct)
....888 313 where r = m= 1. Note that this expansion can also be obtained from (3) but with more involved computations. 2.3 Poissonization Poissonization is a widely used technique in stochastic process, summability of divergent sequence, analysis of algorithms, etc. see, for example, [1, 6, 18, 35, 19]. The idea is roughly described as follows. Given a discrete probability distribution fa k g k0 (or, in general, a complex sequence) consider the Poisson generating function: b( e a j ( 2 C) The usual Poisson heuristic reads: If the sequence fa k g k0 is smooth enough, then a ....
B. C. Berndt, Ramanujan's notebooks, part I, Springer-Verlag, New York, 1985.
Uniform asymptotics of some Abel sums arising in coding theory - Hwang (1997) (Correct)
....appear in a number of applications: n 0; 0 6 k 6 n) 1) where 0 is interpreted as 1. Note that our S n;k di ers from his by 1. For obvious reasons, sums of the type (1) will be referred to as an Abel sum. When k = 0, S n;0 1 is the so called Ramanujan Q function (cf. Berndt [1]) S n;0 1 = Q(n) n j) n ; 2) cf. 28] or Corless et al. 5] The generality of this identity for S n;k , k 1, is given in (12) The Q(n) function was encountered in a number of problems in the analysis of algorithms and combinatorial probability: hashing schemes (Knuth [18] Vitter ....
B. C. Berndt, Ramanujan's notebooks, Part I, Springer-Verlag, New York, 1985.
Uniform asymptotics of some Abel sums arising in coding theory - Hwang (Correct)
....appear in a number of applications: j # 0, 0 where 0 0 is interpreted as 1. Note that our S n,k di#ers from his by 1. For obvious reasons, sums of the type (1) will be referred to as an Abel sum. When k = 0, S n,0 1 is the so called Ramanujan Q function (cf. Berndt [1]) S n,0 1 = Q(n) j) n ; 2) cf. 28] or Corless et al. 5] A general form of this identity for S n,k , k # 1, is given in (12) Preprint submitted to Elsevier Preprint May 19, 1999 The Q(n) function was encountered in a number of problems in the analysis of algorithms and ....
B. C. Berndt, Ramanujan's notebooks, Part I, Springer-Verlag, New York, 1985.
On the Number of Representations of Certain Integers as Sums of.. - Cooper (2002) (Correct)
....sphere j s t j=s 1 (x j 1 2) n. Let r s,t (n) denotes the number of representations of n as a sum of s t squares, of which s are even and t are odd. Then r s,t (n) has the generating function r s,t (n)q (2q) # s,t (q ) z = #(q) x =16q . Then [3] or [24, Ch. 16, Entry 25 (vii) s t) 2 t 4 = # s,t (q) x = #( q) By Jacobi s triple product identity, z = q; q) Therefore (3.1) x) 16q(q (3.2) 256q (3.3) x) 16q( q; q) 16 , ....
....by tan # p = y p x p , 0 # . a(0) 5 4,b(0) 0,c(0) 0,d(0) 1 8,e(0) 1 8 and for n 1, b(n) 2 , otherwise d(n) e(n) if n is even. Proof. From [3] or [24, Ch. 17, Entry 17 (viii) we have 1 4 (2k 1) Therefore a(0) 5 4 and d odd . Next, from [3] or [24, Ch. 17, Entry17 (iii) we have 1 2k . Therefore b(0) 0 and b(n) d odd . For the coe#cients c(n) by ....
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B. C. Berndt, Ramanujan's Notebooks, Part III, Springer-Verlag, New York, 1991.
On the Number of Representations of Certain Integers as Sums of.. - Cooper (2002) (Correct)
....contains only odd powers. Proof. This follows from the Change of Sign Principle [3, p. 126] or [24, Ch. 17, Entry 14 (xii) # Lemma 4.6 ) 264(p 1) z 441(1 14x x = 65520(p 1) Proof. From [3] or [24, Ch. 17, Entries 13 (iii) and (iv) followed by [2] or [24, Ch. 15, Entry 12 (iii) we have ) M(q)N(q) 1 264 . in this is readily seen to be 264(p 1) Also, from [3] or [24, Ch. 17, Entries 13 (iii) and (iv) followed by [2] or [24, Ch. 15, Entry 13 (i) we have (1 14x x 250z (1 x) 441M(q) ....
....= 65520(p 1) Proof. From [3] or [24, Ch. 17, Entries 13 (iii) and (iv) followed by [2] or [24, Ch. 15, Entry 12 (iii) we have ) M(q)N(q) 1 264 . in this is readily seen to be 264(p 1) Also, from [3] or [24, Ch. 17, Entries 13 (iii) and (iv) followed by [2] or [24, Ch. 15, Entry 13 (i) we have (1 14x x 250z (1 x) 441M(q) 250N(q) 691 65520 . in this is readily seen to be 65520(p 1) # Theorem 5.1 If p is an odd prime and H(p) is defined by equation (3.5) then 330 31 352 H(p) Proof. Let ....
B. C. Berndt, Ramanujan's Notebooks, Part II, Springer-Verlag, New York, 1989.
Ramanujan and Euler's Constant - Brent (Correct)
....Riemann zeta function or related functions, they are not convenient for computational purposes. Much of Ramanujan s work was not published during his lifetime, but was summarized in his Notebooks. These were first printed in facsimile [20] and edited editions have since been published by Berndt [4]. Scanning the Notebooks, we find many occurrences of fl. Owing to space limitations, we concentrate on Chapter 4, Entry 9, Corollaries 1 2 [4, I, p. 98] because these are potentially useful for computing fl. Corollary 1 is (in modern notation) 1 X k=1 ( Gamma1) k Gamma1 x k k k = ln x ....
B. C. Berndt, Ramanujan's Notebooks, Parts I--III, Springer-Verlag, New York,
Ramanujan and Euler's Constant - Brent (Correct)
....Riemann zeta function or related functions, they are not convenient for computational purposes. Much of Ramanujan s work was not published during his lifetime, but was summarized in his Notebooks. These were rst printed in facsimile [20] and edited editions have since been published by Berndt [4]. Scanning the Notebooks, we nd many occurrences of . Owing to space limitations, we concentrate on Chapter 4, Entry 9, Corollaries 1 2 [4, I, p. 98] because these are potentially useful for computing . Corollary 1 is (in modern notation) 1 X k=1 ( 1) k 1 x k k k = ln x o(1) 2) ....
B. C. Berndt, Ramanujan's Notebooks, Parts I{III, Springer-Verlag, New York,
The Fourier-Series Method For Inverting Transforms Of.. - Abate, Whitt (1991) (29 citations) (Correct)
....summation formula. With the Poisson summation formula we are able to systematically control the discretization error. The Poisson summation formula itself is a classical result. It is interesting that it has a prominent place in Ramanujan s notebooks; see page 5 and Chapters 13 and 14 of Berndt [11]. The essential idea is to approximate the given function by a periodic function that can be represented by its Fourier series. This explains the name Fourier series method. In the signal processing and time series literature, this is called aliasing; e.g. pp. 26 28 of Rabiner and Gold ....
....Insert Algorithm LATTICE POISSON here (or slightly later) This particular inversion problem is discussed further in 11. Remarks (5.5) There are many formulas essentially equivalent to (5. 38) that can be derived from variants of the Poisson summation formula; e.g. see pages 233 and 235 of Berndt [11]. Elementary algebra shows that p. 233 of Berndt [11] is equivalent to (5.38) and (5.39) To obtain a discrete analog of (5.30) from (3.10a) we can use the following discrete analog of (5.9) k = 1 S m f(u m kp ) e il(u m kp ) 2m k = S a 2km l e i2kmu . 5.40) ....
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B. C. Berndt, Ramanujan's Notebooks, Part II (Springer, New York, 1989).
Euler Sums and Contour Integral Representations - Flajolet, Salvy (1998) (10 citations) (Correct)
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B. C. Berndt, Ramanujan's notebooks, I, Springer, New York, 1989.
Analytica - An Experiment in Combining Theorem Proving and.. - Bauer, Clarke, Zhao (1997) (11 citations) (Correct)
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B. C. Berndt. Ramanujan's Notebooks, Part I. Springer-Verlag, 1985.
Res. Lett. Inf. Math. Sci., 2003, Vol.5, pp 23-59 - Available Online At (Correct)
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B. C. Berndt, Ramanujan's Notebooks, Part V, Springer-Verlag, 1998.
Res. Lett. Inf. Math. Sci., 2003, Vol.5, pp 23-59 - Available Online At (Correct)
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B. C. Berndt, Ramanujan's Notebooks, Part III, Springer-Verlag, 1991.
Res. Lett. Inf. Math. Sci., 2003, Vol.5, pp 11-15 - Available Online At (Correct)
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B. C. Berndt, Ramanujan's Notebooks, Part III, Springer-Verlag, New York, 1991.
Res. Lett. Inf. Math. Sci., 2004, Vol.6, pp157-161 - Available Online At (Correct)
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B. C. Berndt, Ramanujan's Notebooks, Part III, Springer-Verlag, New York, 1991.
Combining symbolic computation and theorem proving: some.. - Edmund Clarke Xudong (1994) (5 citations) (Correct)
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B.C.Berndt, Ramanujan's Notebooks, Part I, Springer-Verlag, 1985, pp 25-43.
Analytica - An Experiment in Combining Theorem Proving and.. - Clarke, Zhao (1992) (6 citations) (Correct)
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B.C.Berndt, Ramanujan's Notebooks, Part I, Springer-Verlag, 1985, pp 25-43. 17
Two-stage allocations and the double Q-function - Agievich (2003) (Correct)
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B. C. Berndt, Ramanujan's notebooks, Part II, Springer-Verlag, Berlin, 1989.
Two-stage allocations and the double Q-function - Agievich (2003) (Correct)
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B. C. Berndt, Ramanujan's notebooks, Part II, Springer-Verlag, Berlin, 1989.
System Description: Analytica 2 - Edmund Clarke Michael (2 citations) (Correct)
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B. C. Berndt. Ramanujan's Notebooks, Part I, pp. 25-43. Springer, 1985.
Singularity Analysis, Hadamard Products, and Tree Recurrences - Fill, Flajolet, Kapur (Correct)
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Berndt, B. C. Ramanujan's Notebooks, Part I. Springer Verlag, 1985.
Singularity Analysis, Hadamard Products, and Tree Recurrences - Fill, Flajolet, Kapur (Correct)
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Berndt, B. C. Ramanujan's Notebooks, Part I. Springer Verlag, 1985.
Residue Theorem and Theta Function Identities - Liu (2001) (Correct)
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B.C. Berndt, Ramanujan's Notebooks, part III, Springer-Verlag, New York, 1991.
Residue Theorem and Theta Function Identities - Liu (2001) (Correct)
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B.C. Berndt, Ramanujan's Notebooks, part II, Springer-Verlag, New York, 1989.
THE RAMANUJAN JOURNAL, 5, 111--128, 2001 c - Appell's Function And (Correct)
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B.C. Berndt, Ramanujan's Notebooks, Part IV. Springer-Verlag, Berlin, 1994.
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