Nikolai Sergeevich Koshlyakov (on the centenary of his birth)

Bogolyubov, N N; Faddeev, Lyudvig Dmitrievich; Ishlinskii, A. Yu.; Koshlyakov, V. N.; Mitropol’skii, Yu. A.

doi:10.1070/rm1990v045n04abeh002382

Cited by 3 publications

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“…The authors of this paper feel that Koshliakov's work has not earned the respect that it deserves in the mathematical community. Some of his best work was achieved under extreme hardship, as these excerpts from a paper written for the centenary of his birth clearly demonstrate [20]. The repressions of the thirties which affected scholars in Leningrad continued even after the outbreak of the Second World War.…”

Section: Introductionmentioning

confidence: 97%

New pathways and connections in number theory and analysis motivated by two incorrect claims of Ramanujan

Berndt

Dixit

Roy

et al. 2017

Advances in Mathematics

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The focus of this paper commences with an examination of three (not obviously related) pages in Ramanujan's lost notebook, pages 336, 335, and 332, in decreasing order of attention. On page 336, Ramanujan proposes two identities, but the formulas are wrong -each is vitiated by divergent series. We concentrate on only one of the two incorrect "identities," which may have been devised to attack the extended divisor problem. We prove here a corrected version of Ramanujan's claim, which contains the convergent series appearing in it. The convergent series in Ramanujan's faulty claim is similar to one used by G. F. Voronoï, G. H. Hardy, and others in their study of the classical Dirichlet divisor problem. This now brings us to page 335, which comprises two formulas featuring doubly infinite series of Bessel functions, the first being conjoined with the classical circle problem initiated by Gauss, and the second being associated with the Dirichlet divisor problem. The first and fourth authors, along with Sun Kim, have written several papers providing proofs of these two difficult formulas in different interpretations. In this monograph, we return to these two formulas and examine them in more general settings.The aforementioned convergent series in Ramanujan's "identity" is also similar to one that appears in a curious identity found in Chapter 15 in Ramanujan's second notebook, written in a more elegant, equivalent formulation on page 332 in the lost notebook. This formula may be regarded as a formula for ζ( 1 2 ), and in 1925, S. Wigert obtained a generalization giving a formula for ζ( 1 k ) for any even integer k ≥ 2. We extend the work of Ramanujan and Wigert in this paper.The Voronoï summation formula appears prominently in our study. In particular, we generalize work of J. R. Wilton and derive an analogue involving the sum of divisors function σs(n).The modified Bessel functions Ks(x) arise in several contexts, as do Lommel functions. We establish here new series and integral identities involving modified Bessel functions and modified Lommel functions. Among other results, we establish a modular transformation for an infinite series involving σs(n) and modified Lommel functions. We also discuss certain obscure related work of N. S. Koshliakov. We define and discuss two new related classes of integral transforms, which we call Koshliakov transforms, because he first found elegant special cases of each.

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Section: Introductionmentioning

confidence: 97%

New pathways and connections in number theory and analysis motivated by two incorrect claims of Ramanujan

Berndt

Dixit

Roy

et al. 2017

Advances in Mathematics

View full text Add to dashboard Cite

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Lyudvig Dmitrievich Faddeev (on his sixtieth birthday)

Osipov¹,

Gonchar²,

Новиков³

et al. 1995

Russ. Math. Surv.

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A review of hybrid integral transform solutions in fluid flow problems with heat or mass transfer and under Navier–Stokes equations formulation

Cotta

Lisbôa

Curi

et al. 2019

Numerical Heat Transfer, Part B: Fundamentals

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The Generalized Integral Transform Technique (GITT) is reviewed as a hybrid numericalanalytical approach for fluid flow problems, with or without heat and mass transfer, here with emphasis on the literature related to flow problems formulated through the full Navier-Stokes equations. A brief overview of the integral transform methodology is first provided for a general nonlinear convection-diffusion problem. Then, different alternatives of eigenfunction expansion strategies are discussed in the integral transformation of problems for which the fluid flow model is either based on the primitive variables or the streamfunction-only formulations, as applied to both steady and transient states. Representative test cases are selected to illustrate the different eigenfunction expansion approaches, with convergence being analyzed for each situation. In addition, fully converged integral transform results are critically compared to previously reported simulations obtained from traditional purely discrete methods.

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Nikolai Sergeevich Koshlyakov (on the centenary of his birth)

Cited by 3 publications

References 0 publications

New pathways and connections in number theory and analysis motivated by two incorrect claims of Ramanujan

New pathways and connections in number theory and analysis motivated by two incorrect claims of Ramanujan

Lyudvig Dmitrievich Faddeev (on his sixtieth birthday)

A review of hybrid integral transform solutions in fluid flow problems with heat or mass transfer and under Navier–Stokes equations formulation

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