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Journal articles on the topic 'Lerch's theorem'

Author: Grafiati

Published: 4 June 2021

Last updated: 14 February 2022

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1

Dolník, Matej, and Tomáš Kisela. "Lerch’s theorem on nabla time scales." Mathematica Slovaca 69, no. 5 (2019): 1127–36. http://dx.doi.org/10.1515/ms-2017-0295.

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Abstract The paper discusses uniqueness of Laplace transform considered on nabla time scales. As the main result, a nabla time scales analogue of Lerch’s theorem ensuring uniqueness of Laplace image is proved for so-called simply periodic time scales. Moreover, several presented counterexamples demonstrate that the uniqueness of Laplace image does not occur on general time scales when the nabla approach is employed. Other special properties of Laplace transform on nabla time scales, such as potential disconnectedness of domain of convergence, are addressed as well.

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2

WANG, LI-YUAN, and HAI-LIANG WU. "APPLICATIONS OF LERCH’S THEOREM TO PERMUTATIONS OF QUADRATIC RESIDUES." Bulletin of the Australian Mathematical Society 100, no. 3 (2019): 362–71. http://dx.doi.org/10.1017/s000497271900073x.

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Let $n$ be a positive integer and $a$ an integer prime to $n$. Multiplication by $a$ induces a permutation over $\mathbb{Z}/n\mathbb{Z}=\{\overline{0},\overline{1},\ldots ,\overline{n-1}\}$. Lerch’s theorem gives the sign of this permutation. We explore some applications of Lerch’s result to permutation problems involving quadratic residues modulo $p$ and confirm some conjectures posed by Sun [‘Quadratic residues and related permutations and identities’, Preprint, 2018, arXiv:1809.07766]. We also study permutations involving arbitrary $k$th power residues modulo $p$ and primitive roots modulo a power of $p$.

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3

Barnes, K. J. "Avoiding the theorem of Lerche and Shore." Physics Letters B 468, no. 1-2 (1999): 81–85. http://dx.doi.org/10.1016/s0370-2693(99)00865-5.

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4

Tzermias, Pavlos. "A note on a paper by Brenner." International Journal of Mathematics and Mathematical Sciences 31, no. 11 (2002): 701–2. http://dx.doi.org/10.1155/s0161171202202197.

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5

Laurinčikas, Antanas, and Renata Macaitienė˙. "Joint universality of the Riemann zeta-function and Lerch zeta-functions." Nonlinear Analysis: Modelling and Control 18, no. 3 (2013): 314–26. http://dx.doi.org/10.15388/na.18.3.14012.

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In the paper, we prove a joint universality theorem for the Riemann zeta-function and a collection of Lerch zeta-functions with parameters algebraically independent over the field of rational numbers.

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6

Laurinčikas, Antanas, and Kohji Matsumoto. "The joint universality and the functional independence for Lerch zeta-functions." Nagoya Mathematical Journal 157 (2000): 211–27. http://dx.doi.org/10.1017/s002776300000725x.

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The joint universality theorem for Lerch zeta-functions L(λl, αl, s) (1 ≤ l ≤ n) is proved, in the case when λls are rational numbers and αls are transcendental numbers. The case n = 1 was known before ([12]); the rationality of λls is used to establish the theorem for the “joint” case n ≥ 2. As a corollary, the joint functional independence for those functions is shown.

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7

Kutbi, M. A., and A. A. Attiya. "Differential Subordination Results for Certain Integrodifferential Operator and Its Applications." Abstract and Applied Analysis 2012 (2012): 1–13. http://dx.doi.org/10.1155/2012/638234.

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We introduce an integrodifferential operatorJs,b(f) which plays an important role in theGeometric Function Theory. Some theorems in differential subordination forJs,b(f) are used. Applications inAnalytic Number Theoryare also obtained which give new results for Hurwitz-Lerch Zeta function and Polylogarithmic function.

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8

Wang, Xiao-Yuan, Lei Shi, and Zhi-Ren Wang. "Certain Integral Operator Related to the Hurwitz–Lerch Zeta Function." Journal of Complex Analysis 2018 (April 8, 2018): 1–7. http://dx.doi.org/10.1155/2018/5915864.

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The aim of the present paper is to investigate several third-order differential subordinations, differential superordination properties, and sandwich-type theorems of an integral operator Ws,bf(z) involving the Hurwitz–Lerch Zeta function. We make some applications of the operator Ws,bf(z) for meromorphic functions.

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9

Laurinčikas, A., and K. Matsumoto. "Joint value-distribution theorems on Lerch zeta-functions. II." Lithuanian Mathematical Journal 46, no. 3 (2006): 271–86. http://dx.doi.org/10.1007/s10986-006-0027-x.

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10

Laurincikas, A., and Kohji Matsumoto. "Joint value-distribution theorems for the Lerch zeta-functions." Lithuanian Mathematical Journal 38, no. 3 (1998): 238–49. http://dx.doi.org/10.1007/bf02465899.

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11

Laurinčikas, A. "On the effectivization of the universality theorem for the Lerch zeta-function." Lithuanian Mathematical Journal 40, no. 2 (2000): 135–39. http://dx.doi.org/10.1007/bf02467152.

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12

Srivastava, H. M., та Sébastien Gaboury. "New Expansion Formulas for a Family of theλ-Generalized Hurwitz-Lerch Zeta Functions". International Journal of Mathematics and Mathematical Sciences 2014 (2014): 1–13. http://dx.doi.org/10.1155/2014/131067.

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We derive several new expansion formulas for a new family of theλ-generalized Hurwitz-Lerch zeta functions which were introduced by Srivastava (2014). These expansion formulas are obtained by making use of some important fractional calculus theorems such as the generalized Leibniz rules, the Taylor-like expansions in terms of different functions, and the generalized chain rule. Several (known or new) special cases are also considered.

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13

Laurinčikas, A. "A limit theorem for the Lerch zeta-function in the space of analytic functions." Lithuanian Mathematical Journal 37, no. 2 (1997): 146–55. http://dx.doi.org/10.1007/bf02465887.

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14

Chen, Bin. "On the dual nature theory of bilateral series associated to mock theta functions." International Journal of Number Theory 14, no. 01 (2017): 63–94. http://dx.doi.org/10.1142/s1793042118500069.

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In recent work, Hickerson and Mortenson introduced a dual notion between Appell–Lerch sums and partial theta functions. In this sense, Appell–Lerch sums and partial theta functions appear to be dual to each other. In this paper, by making the substitution [Formula: see text] in the tail of the associated bilateral series of mock theta functions and universal mock theta functions, we demonstrate how to obtain duals of the second type in terms of Appell–Lerch sums defined by Mortenson for such functions. Then by using the substitution [Formula: see text] in duals of the second type of each bilateral series, we present how to translate between identities expressing [Formula: see text]-hypergeometric series in terms of Appell–Lerch sums and identities expressing [Formula: see text]-hypergeometric series in terms of partial theta functions. Indeed, we obtain only four duals in terms of partial theta functions of duals of the second type in terms of Appell–Lerch sums of bilateral series associated to mock theta functions. As an application, we construct Ramanujan radial limits by using these bilateral series with mock modular behavior in terms of Appell–Lerch sums for some order mock theta functions. This method is well-suited for the other order mock theta functions.

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15

Hu, Qiuxia, Hanfei Song, and Zhizheng Zhang. "Third-order mock theta functions." International Journal of Number Theory 16, no. 01 (2019): 91–106. http://dx.doi.org/10.1142/s1793042120500050.

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In [G. E. Andrews and B. C. Berndt, Ramanujan’s Lost Notebook, Part II (Springer, New York, 2009), Entry 3.4.7, p. 67; Y.-S. Choi, The basic bilateral hypergeometric series and the mock theta functions, Ramanujan J. 24(3) (2011) 345–386; B. Chen, Mock theta functions and Appell–Lerch sums, J. Inequal Appl. 2018(1) (2018) 156; E. Mortenson, Ramanujan’s radial limits and mixed mock modular bilateral [Formula: see text]-hypergeometric series, Proc. Edinb. Math. Soc. 59(3) (2016) 1–13; W. Zudilin, On three theorems of Folsom, Ono and Rhoades, Proc. Amer. Math. Soc. 143(4) (2015) 1471–1476], the authors found the bilateral series for the universal mock theta function [Formula: see text]. In [Choi, 2011], the author presented the bilateral series connected with the odd-order mock theta functions in terms of Appell–Lerch sums. However, the author only derived the associated bilateral series for the fifth-order mock theta functions. The purpose of this paper is to further derive different types of bilateral series for the third-order mock theta functions. As applications, the identities between the two-group bilateral series are obtained and the bilateral series associated to the third-order mock theta functions are in fact modular forms. Then, we consider duals of the second type in terms of Appell–Lerch sums and duals in terms of partial theta functions defined by Hickerson and Mortenson of duals of the second type in terms of Appell–Lerch sums of such bilateral series associated to some third-order mock theta functions that Chen did not discuss in [On the dual nature theory of bilateral series associated to mock theta functions, Int. J. Number Theory 14 (2018) 63–94].

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16

Chang, Ching-Hua, and Chung-Wei Ha. "A multiplication theorem for the Lerch zeta function and explicit representations of the Bernoulli and Euler polynomials." Journal of Mathematical Analysis and Applications 315, no. 2 (2006): 758–67. http://dx.doi.org/10.1016/j.jmaa.2005.08.013.

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17

Wiser, Antonin. "Nachtigall oder Lerche: Übersetzen ohne Light-Bild." Recherches germaniques, no. 49 (December 12, 2019): 139–50. http://dx.doi.org/10.4000/rg.2691.

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18

Chen, Shi-Chao. "Vanishing of the coefficients of a half Lerch sum." International Journal of Number Theory 15, no. 02 (2019): 251–63. http://dx.doi.org/10.1142/s1793042119500106.

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We consider the vanishing problem on a [Formula: see text]-series related to Lerch sum and the first moment of crank partitions and overpartitions. We prove that almost all of the coefficients of this series vanish. We also bound the small values of the coefficients, which improve recent work of Xiong.

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19

Mizuno, Yoshinori. "Generalized Lerch formulas: Examples of zeta-regularized products." Journal of Number Theory 118, no. 2 (2006): 155–71. http://dx.doi.org/10.1016/j.jnt.2005.08.005.

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20

Murty, M. Ram, and Siddhi Pathak. "Convolution of values of the Lerch zeta-function." Journal of Number Theory 217 (December 2020): 1–22. http://dx.doi.org/10.1016/j.jnt.2020.01.008.

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21

Nakamura, Takashi. "Some formulas related to Hurwitz–Lerch zeta functions." Ramanujan Journal 21, no. 3 (2010): 285–302. http://dx.doi.org/10.1007/s11139-009-9199-4.

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22

Jorgenson, Jay, and Serge Lang. "Hilbert-Asai Eisenstein series, regularized products, and heat kernels." Nagoya Mathematical Journal 153 (1999): 155–88. http://dx.doi.org/10.1017/s0027763000006942.

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AbstractIn a famous paper, Asai indicated how to develop a theory of Eisenstein series for arbitrary number fields, using hyperbolic 3-space to take care of the complex places. Unfortunately he limited himself to class number 1. The present paper gives a detailed exposition of the general case, to be used for many applications. First, it is shown that the Eisenstein series satisfy the authors’ definition of regularized products satisfying the generalized Lerch formula, and the basic axioms which allow the systematic development of the authors’ theory, including the Cramér theorem. It is indicated how previous results of Efrat and Zograf for the strict Hilbert modular case extend to arbitrary number fields, for instance a spectral decomposition of the heat kernel periodized with respect to SL2 of the integers of the number field. This gives rise to a theta inversion formula, to which the authors’ Gauss transform can be applied. In addition, the Eisenstein series can be twisted with the heat kernel, thus encoding an infinite amount of spectral information in one item coming from heat Eisenstein series. The main expected spectral formula is stated, but a complete exposition would require a substantial amount of space, and is currently under consideration.

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23

Xiong, Xinhua. "Small values of coefficients of a half Lerch sum." International Journal of Number Theory 13, no. 09 (2017): 2461–70. http://dx.doi.org/10.1142/s1793042117501366.

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Andrews, Dyson and Hickerson proved many interesting properties of coefficients for a Ramanujan’s [Formula: see text]-hypergeometric series by relating it to real quadratic field [Formula: see text] and using the arithmetic of [Formula: see text] to solve a conjecture of Andrews on the distributions of its Fourier coefficients. Motivated by Andrews’s conjecture, we discuss an interesting [Formula: see text]-hypergeometric series which comes from a Lerch sum and rank and crank moments for partitions and overpartitions. We give Andrews-like conjectures for its coefficients. We obtain partial results on the distributions of small values of its coefficients toward these conjectures.

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24

Bringmann, Kathrin, Amanda Folsom, and Ken Ono. "q-series and weight 3/2 Maass forms." Compositio Mathematica 145, no. 03 (2009): 541–52. http://dx.doi.org/10.1112/s0010437x09004072.

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AbstractDespite the presence of many famous examples, the precise interplay between basic hypergeometric series and modular forms remains a mystery. We consider this problem for canonical spaces of weight 3/2 harmonic Maass forms. Using recent work of Zwegers, we exhibit forms that have the property that their holomorphic parts arise from Lerch-type series, which in turn may be formulated in terms of the Rogers–Fine basic hypergeometric series.

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25

Baruah, Nayandeep Deka, and Nilufar Mana Begum. "Proofs of some conjectures of Chan on Appell–Lerch sums." Ramanujan Journal 51, no. 1 (2019): 99–115. http://dx.doi.org/10.1007/s11139-018-0076-x.

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26

VON WILDERT, GERO. "Anton Lerch – geduppelt? Zum sogenannten "Doppelgänger" in Hofmannsthals Reitergeschichte." Seminar: A Journal of Germanic Studies 29, no. 2 (1993): 125–37. http://dx.doi.org/10.3138/sem.v29.2.125.

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27

Reynolds, Robert, and Allan Stauffer. "A Note on the Definite Integral of the Lerch Function." European Journal of Pure and Applied Mathematics 14, no. 3 (2021): 788–802. http://dx.doi.org/10.29020/nybg.ejpam.v14i3.4030.

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This is a compilation of definite integrals involving the Lerch, Polylogarithm, Hurwitz functions and fundamental constants. Connections to previous work is compared and discussed. This collection of definite integrals is new in current literature.

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28

Reynolds, Robert, and Allan Stauffer. "Mellin Transform of Logarithm and Quotient Function with Reducible Quartic Polynomial in Terms of the Lerch Function." Axioms 10, no. 3 (2021): 236. http://dx.doi.org/10.3390/axioms10030236.

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A class of definite integrals involving a quotient function with a reducible polynomial, logarithm and nested logarithm functions are derived with a possible connection to contact problems for a wedge. The derivations are expressed in terms of the Lerch function. Special cases are also derived in terms fundamental constants. The majority of the results in this work are new.

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29

Olivetto, René. "On the Fourier coefficients of meromorphic Jacobi forms." International Journal of Number Theory 10, no. 06 (2014): 1519–40. http://dx.doi.org/10.1142/s1793042114500419.

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In this paper, we describe the automorphic properties of the Fourier coefficients of meromorphic Jacobi forms. Extending results of Dabholkar, Murthy, and Zagier, and Bringmann and Folsom, we prove that the canonical Fourier coefficients of a meromorphic Jacobi form φ(z; τ) are the holomorphic parts of some (vector-valued) almost harmonic Maass forms. We also give a precise description of their completions, which turn out to be uniquely determined by the Laurent coefficients of φ at each pole, as well as some well-known real analytic functions, that appear for instance in the completion of Appell–Lerch sums.

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30

Kim, Namhoon. "Ramanujan's integral identities of the Riemann Ξ-function and the Lerch transcendent". Journal of Number Theory 168 (листопад 2016): 292–305. http://dx.doi.org/10.1016/j.jnt.2016.04.006.

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31

McIntosh, Richard J. "Some identities for Appell–Lerch sums and a universal mock theta function." Ramanujan Journal 45, no. 3 (2017): 767–79. http://dx.doi.org/10.1007/s11139-016-9869-y.

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32

Nagoshi, Hirofumi. "Joint universality and simple a-points of Lerch zeta-functions." Moscow Journal of Combinatorics and Number Theory 10, no. 2 (2021): 121–39. http://dx.doi.org/10.2140/moscow.2021.10.121.

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33

Srivastava, H. M., Dragana Jankov, Tibor K. Pogány, and R. K. Saxena. "Two-sided inequalities for the extended Hurwitz–Lerch Zeta function." Computers & Mathematics with Applications 62, no. 1 (2011): 516–22. http://dx.doi.org/10.1016/j.camwa.2011.05.035.

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34

Nakamura, Takashi. "Applications of inversion formulas to the joint t-universality of Lerch zeta functions." Journal of Number Theory 123, no. 1 (2007): 1–9. http://dx.doi.org/10.1016/j.jnt.2006.05.012.

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35

Nakamura, Takashi. "The existence and the non-existence of joint t-universality for Lerch zeta functions." Journal of Number Theory 125, no. 2 (2007): 424–41. http://dx.doi.org/10.1016/j.jnt.2006.12.008.

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36

DEZANI-CIANCAGLINI, MARIANGIOLA, GIUSEPPE LONGO, and JONATHAN P. SELDIN. "Preface." Mathematical Structures in Computer Science 9, no. 4 (1999): 321. http://dx.doi.org/10.1017/s0960129598009682.

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This special double issue of Mathematical Structures in Computer Science is in honour of Roger Hindley and is devoted to the topic of lambda-calculus and logic.It is a great pleasure for us to greet Roger Hindley on the occasion of his retirement from the University of Wales, Swansea, and his 60th birthday. We have known Roger for many years and we have had the chance to collaborate with him and appreciate his intellectual standard, his remarkable mathematical rigor, and his inexhaustible sense of humour. This has enabled Roger to step back critically even in the face of a difficult mathematical task and help to solve it by a new way of looking at it.Roger Hindley's dissertation concerned the Church–Rosser Theorem and was a significant contribution to the topic. His subsequent work spanned many aspects of lambda-calculus, covering both its models and applications. To mention just a few, he produced work on axioms for Curry's strong (eta) reduction, comparing lambda and combinatory reductions (and models), models for type assignment, and formulas as types for some nonstandard systems (intersection types, BCK systems, etc.).Roger Hindley collaborated with Jonathan Seldin on two well-known introductory books on the subject (Bruce Lercher also collaborated as an author on the first of these). More recently, he has published an introduction to type assignment. He was also co-author with H. B. Curry and J. Seldin on Combinatory Logic, vol. II, which is an important research publication on the subject.Roger has played an important role in the lambda-calculus community over the years as that community has grown; in particular, he has been an active organiser of many conferences on the topic. In fact, his success in disseminating knowledge about the lambda calculus, particularly in the United Kingdom, means that Roger may be considered a ‘Godfather’ of ML and its type system.(In preparing this special issue of Mathematical Structures in Computer Science, we have been fortunate enough to receive too many excellent papers for one double issue. As a result, additional papers by colleagues who wish to honour Roger will appear in future issues of this journal.)

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37

Reynolds, Robert, and Allan Stauffer. "Definite Integral of Power and Algebraic Functions in terms of the Lerch Function." European Journal of Pure and Applied Mathematics 14, no. 3 (2021): 980–88. http://dx.doi.org/10.29020/nybg.ejpam.v14i3.4017.

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Bierens de haan (1867) evaluated a definite integral involving the cotangent function and this result was also listed in Gradshteyn and Ryzhik (2007). The objective of this present note is to use this integral along with Cauchy's integral formula to derive a definite logarithmic integral in terms of the Lerch function. We will use this integral formula to produce a table of known and new results in terms of special functions and thereby expanding the list of definite integrals in both text books.

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38

Srivastava, Hari M., Sébastien Gaboury, and Abdelmejid Bayad. "Expansion formulas for an extended Hurwitz-Lerch zeta function obtained via fractional calculus." Advances in Difference Equations 2014, no. 1 (2014): 169. http://dx.doi.org/10.1186/1687-1847-2014-169.

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39

Srivastava, H. M., Recep Sahin, and Oguz Yagci. "A family of incomplete Hurwitz-Lerch zeta functions of two variables." Miskolc Mathematical Notes 21, no. 1 (2020): 401. http://dx.doi.org/10.18514/mmn.2020.3059.

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40

Reynolds, Robert, and Allan Stauffer. "Note on a Stieltjes Transform in terms of the Lerch Function." European Journal of Pure and Applied Mathematics 14, no. 3 (2021): 723–36. http://dx.doi.org/10.29020/nybg.ejpam.v14i3.3991.

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In this work the authors derive the Stieltjes transform of the logarithmic function in terms of the Lerch function. This transform is used to derive closed form solutions involving fundamental constants and special functions. Specifically we derive the definite integral given by\[\int_{0}^{\infty} \frac{(1-b x)^m \log ^k(c (1-b x))+(b x+1)^m \log ^k(c (b x+1))}{a+x^2}dx\]where $a,b,c,m$ and $k$ are general complex numbers subject to the restrictions given in connection with the formulas.

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41

Reynolds, Robert, and Allan Stauffer. "The Double Laplace Transform Expressed in terms of the Lerch Transcendent." European Journal of Pure and Applied Mathematics 14, no. 3 (2021): 618–37. http://dx.doi.org/10.29020/nybg.ejpam.v14i3.3987.

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In this manuscript, the authors derive a formula for the double Laplace transform expressed in terms of the Lerch Transcendent. The log term mixes the variables so that the integral is not separable except for special values of k. The method of proof follows the method used by us to evaluate single integrals. This transform is then used to derive definite integrals in terms of fundamental constants, elementary and special functions. A summary of the results is produced in the form of a table of definite integrals for easy referencing by readers.

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42

Cvijović, Djurdje. "Exponential and trigonometric sums associated with the Lerch zeta and Legendre chi functions." Computers & Mathematics with Applications 59, no. 4 (2010): 1484–90. http://dx.doi.org/10.1016/j.camwa.2010.01.026.

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43

Srivastava, H. M. "A New Family of the λ -Generalized Hurwitz-Lerch Zeta Functions with Applications". Applied Mathematics & Information Sciences 8, № 4 (2014): 1485–500. http://dx.doi.org/10.12785/amis/080402.

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44

Srivastava, H. M., Nidhi Jolly, Manish Kumar Bansal та Rashmi Jain. "A new integral transform associated with the $$\lambda $$ λ -extended Hurwitz–Lerch zeta function". Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 113, № 3 (2018): 1679–92. http://dx.doi.org/10.1007/s13398-018-0570-4.

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45

Meemark, Yotsanan, and Sirawich Chinwarakorn. "Lerch's Theorems over Function Fields." Integers 10, no. 1 (2010). http://dx.doi.org/10.1515/integ.2010.004.

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46

Genienė, Danutė Regina. "A two-dimensional limit theorem for Lerch zeta-functions. II." Lietuvos matematikos rinkinys 52 (December 15, 2011). http://dx.doi.org/10.15388/lmr.2011.al02.

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47

Skula, Ladislav. "A note on some relations among special sums of reciprocals modulo p." Mathematica Slovaca 58, no. 1 (2008). http://dx.doi.org/10.2478/s12175-007-0050-3.

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AbstractIn this note the sums s(k, N) of reciprocals $$\sum\limits_{\tfrac{{kp}}{N} < x < \tfrac{{(k + 1)p}}{N}} {\tfrac{1}{x}(mod p)} $$ are investigated, where p is an odd prime, N, k are integers, p does not divide N, N ≥ 1 and 0 ≤ k ≤ N − 1. Some linear relations for these sums are derived using “logarithmic property” and Lerch’s Theorem on the Fermat quotient. Particularly in case N = 10 another linear relation is shown by means of Williams’ congruences for the Fibonacci numbers.

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48

Genienė, Danutė. "The Lerch zeta-function with algebraic irrational parameter." Lietuvos matematikos rinkinys 50 (December 20, 2009). http://dx.doi.org/10.15388/lmr.2009.01.

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49

Agarwal, P., S. Kanemitsu, and T. Kuzumaki. "On some Lambert-like series." Hardy-Ramanujan Journal Volume 42 - Special... (May 20, 2020). http://dx.doi.org/10.46298/hrj.2020.6461.

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International audience In this note, we study radial limits of power and Laurent series which are related to the Lerch zeta-function or polylogarithm function. As has been pointed out in [CKK18], there have appeared many instances in which the imaginary part of the Lerch zeta-function was considered by eliminating the real part by considering the odd part only. Mordell studied the properties of the power series resembling Lambert series, and in particular considered whether the limit function is a rational function or not. Our main result is the elucidation of the threshold case of b_n = 1/n studied by Mordell [Mor63], revealing that his result is the odd part of Theorem 1.1 in view of the identities (1.9), (1.5). We also refer to Lambert series considered by Titchmarsh [Tit38] in connection with Estermann's zeta-functions.

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Bradley-Thrush, J. G. "Properties of the Appell–Lerch function (I)." Ramanujan Journal, September 3, 2021. http://dx.doi.org/10.1007/s11139-021-00445-4.

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