All the trinomial roots, their powers and logarithms from the Lambert series, Bell polynomials and Fox–Wright function: illustration for genome multiplicity in survival of irradiated cells
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Abstract
All the roots of the general nth degree trinomial admit certain convenient representations in terms of the Lambert and Euler series for the asymmetric and symmetric cases of the trinomial equation, respectively. Previously, various methods have been used to provide the proofs for the general terms of these two series. Taking n to be any real or complex number, we presently give an alternative proof using the Bell (or exponential) polynomials. The ensuing series is summed up yielding a single, compact, explicit, analytical formula for all the trinomial roots as the confluent Fox–Wright function \({}_1\Psi _1\). Moreover, we also derive a slightly different, single formula of the trinomial root raised to any power (real or complex number) as another \({}_1\Psi _1\) function. Further, in this study, the logarithm of the trinomial root is likewise expressed through a single, concise series with the binomial expansion coefficients or the Pochhammer symbols. These findings are anticipated to be of considerable help in various applications of trinomial roots. Namely, several properties of the \({}_1\Psi _1\) function can advantageously be employed for its implementations in practice. For example, the simple expressions for the asymptotic limits of the \({}_1\Psi _1\) function at both small and large values of the independent variable can be used to readily predict, by analytical means, the critical behaviors of the studied system in the two extreme conditions. Such limiting situations can be e.g. at the beginning of the time evolution of a system, and in the distant future, if the independent variable is time, or at low and high doses when the independent variable is radiation dose, etc. The present analytical solutions for the trinomial roots are numerically illustrated in the genome multiplicity corrections for survival of synchronous cell populations after irradiation.
Keywords
Trinomial roots Trinomial equations Lambert functions Euler series1 Introduction
Since the topics of the trinomial roots and the Lambert function have historically been tightly intertwined, we shall subdivide this introductory section into two parts, one dealing with the former and the other with the latter subject.
1.1 Trinomial roots
The theme of the roots of trinomials has a remarkable history beginning with Lambert in 1758 [1, 2], followed by Euler in 1777 [3, 4] and continued by many authors during the past 260 years to the present. In particular, it is from finding all the trinomial roots that the important subject of the Lambert W and Euler T functions emerged in the literature. Research on trinomial roots resulted in numerous reports, some of which are given in Refs. [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52] (1851–2018). Presently, we primarily focus upon derivations of the analytical formulae for all the roots of trinomials through the series developments using the Bell polynomials [53] (1934) and the Fox–Wright function [54, 55, 56, 57, 58] (1933–1961). The Bell (or exponential) polynomials arise in obtaining the closed expressions for general derivatives of functions. For example, the Faà di Bruno formula [59, 60, 61] (1855–2002) for the nth derivative of composite functions can be derived by using the Bell polynomials, as shown by Riordan [62, 63] (1946,1978). The Fox–Wright function \({}_n\Psi _m\) is an extension of the generalized Gauss hypergeometric function \({}_nF_m.\) The confluent Fox–Wright function \({}_1\Psi _1\) is the generalized Kummer confluent hypergeometric function \({}_1F_1\) . While the series for \({}_1F_1\) in powers of its independent variable (say x) is known to converge at any finite \(x\, (x <\infty ),\) the corresponding series for \({}_1\Psi _1\) in x converges only within its convergence radius \(R\, (x < R).\)
In 1777, Euler [3] found a series for all the roots of the symmetrized form of the trinomial characteristic equation. Subsequently, over a long period of time, using various methods, the Euler formula has been proven by a number of authors ranging from McClintock in 1895 [11] to Wang in 2016 [50]. We presently give yet another proof of the Euler formula for all the trinomial roots by deriving the explicit expression for the general expansion coefficient in terms of the complete Bell polynomials \(B_n.\) Moreover, transforming these multivariate to univariate polynomials, the expansion coefficients are reduced to the binomial coefficients and the Pochhammer symbols \((a)_n.\) Finally, the obtained series is explicitly summed up with the result given by the confluent Fox–Wright \({}_1\Psi _1\) function.
The Fox–Wright functions \({}_n\Psi _m\) [54, 55, 56, 57, 58] and its generalizations have been used in a number of studies on different subjects and a few articles are listed in Refs. [45, 46, 47, 64, 65, 66, 67] (1994–2007). The usefulness of the analytical formula for trinomial roots in terms of the confluent Fox–Wright function \({}_1\Psi _1\) is in the possibility to exploit the known asymptotic behaviors of the \({}_1\Psi _1\) function at both small and large values of its independent variable x. This is exemplified in the present illustration of the trinomial roots encountered in a radiobiological model for cell survival after exposure to radiation.
1.2 The Lambert W and Euler T function
The Lambert and Euler functions, with their most frequently encountered properties, have thoroughly been reviewed in the literature. Therefore, all that is given in this subsection is mainly a complement to the existing compilations of the bibliography on this subject matter. Despite numerous entries in the cited publications, the present list of references is still far from being exhaustive due to a huge number of reported studies. Because of the versatile nature of applications of these two functions in various disciplines, we will categorize the selected articles according to their research branches.
The Lambert W function [1, 2] and the related Euler T function [3, 4] play a very important role across interdisciplinary research. These two functions are the multivalued solutions of the transcendental equations \(y=x\mathrm{e}^x\, [\therefore \,\, x=W(y)]\) and functions \(y=x\mathrm{e}^{x}\, [\therefore \,\,x=T(y)].\) They arise from a linearexponential, or equivalently, linearlogarithmic equations for the unknown, sought quantity. This special combination of the two elementary functions describes two different behavioral patterns (linear and exponential or linear and logarithmic) that a large number of phenomena share in vastly different fields. The underlying common physical, chemical or biological effects behind a linkage of a linear with an exponential term is often related to two different stages of a complete process of timeevolution of a generic dynamical system. These stages might compete with each other, or they could correspond to a slow and a fast component of the whole developmental process, or they could be associated with the two complementary mechanisms, etc. Such two components may characterize e.g. the rise and fall of the studied observables (experimentally measurable quantities) that describe the behavior of a system in varying environmental conditions under the influence of an external agent. For example, a system of coupled differential rate equations from chemical kinetics (that cannot be solved exactly by analytical means) can be approximately reduced (within a quasistationary state assumption) to a linearexponential or linearlogarithmic transcendental equation whose exact solution is the Lambert function. This occurs in the MichaelisMenten formalism [68] (1913) for enzyme catalysis in the Briggs–Haldane setting [69] (1925). The same linearexponential pattern behavior is routinely encountered in many systems whose time evolution obeys differential or difference equations. Such time evolution is often accompanied with time delays, in which case the delayed differential equations are used, and these end up with a linearexponential transcendental equation which yields exactly the Lambert function.
Of course, these transcendental equations can be solved by numerical means (e.g. by the Newton iteration). However, the possibility of obtaining the exact analytical solution of such equations, e.g. through the Lambert function, is appealing. The reason is that a closed, analytical form of a function is invaluable as it provides the necessary asymptotic forms both at small and large values of the independent variable. Such asymptotes govern the development of the system in the two extreme conditions and provide a way to control and, indeed, predict the behavioral patterns. In the last 20 years, the interdisciplinary literature witnessed an ever increasing interest in the Lambert function. It is anticipated that this enviable trend will be pursued in the next 20 years and beyond.
The mentioned circumstances embodied through the linearexponential mathematical form in the transcendental equations are ubiquitous and this is the main reason for the universal applicability of the Lambert function in distant and seemingly unrelated fields. It would be virtually impossible to enumerate various mechanisms in versatile research branches that could produce the Lambert function as the end result. The number of articles dealing with this remarkable function is enormous, and no review can be exhaustive enough in citing and/or commenting on a greater part the related publications. The present work is no exception, and we shall content ourselves to mention only a smaller fraction of the past contributions to this topic. What makes an investigative result important is its usefulness to a wider circle of other researchers over an extended period of time. The Lambert function passed this test of time as testified by an unprecedented use of this function in mathematics, physics, astrophysics, chemistry, biology, medicine, population genetics, ecology, sociology, education, energetics, technology, etc. To help the general reader (with a hope of motivating a further extension of the applications of the Lambert function) and especially due to an unprecedentedly abundant literature, it is deemed instructive to group the publications into several categories.
The first quoted are the originators, Lambert and Euler, with two cited articles per author. Subsequently, general information is collected by quoting books, tabular publications, PhD Theses, reviews, international workshops, websites and posters. This is followed by quoting computational contributions (algorithms, programs, libraries, open source codes) and articles with ceveral quite accurate approximate formulae for the Lambert function.
The next quoted are the publications on the applications of the Lambert and Euler functions in various disciplines, such as mathematics, physics, astrophysics/astronomy, chemistry, biomedicine, ecology, sociology, technology and education. Some of these publications deal exclusively with the Lambert and Euler functions, whereas the other studies address a number of features of these functions among the other treated topics.

Lambert’s articles On the series solution for trinomial roots [1, 2](1758, 1770).

Euler’s articles On the Lambert series for trinomial roots [3, 4] (1777, 1783).

Books Series solutions of algebraic equations, theory of transcendental functions, enumerative combinatorics, population of species, timedelayed systems, etc. [70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83] (1906–2016). In particular, Pólya and Szegő [71] (1925) examined the function \(y=x\mathrm{e}^x\) and found its inverse. Their solution is recognized as the Lambert function, whose contemporary notation is W and, therefore, the inverse of \(y=x\mathrm{e}^x\) from Ref. [71] is given by \(x=W(y).\)

Tables Tables of mathematical properties of the Lambert W function and their integrals: [84, 85] (2004, 2010). The former study is in Russian and the latter work is from the American National Institute of Standards and Technology (NIST).

Ph.D. Theses Linear timedelayed systems, growth models for plants, etc [86, 87, 88, 89, 90] (2007–2012).

Reviews Asymptotic behaviors, links to trinomial zeros, solar cells, biochemical kinetics, enzyme catalysis, radiobiological models for radiotherapy in medicine, ecology and evolution, etc. [91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105] (1996–2018).

Conferences A workshop marking the first 20 years of a revitalization of the Lambert function, a meeting on the Lambert function alongside some other special functions in optimization [106, 107] (2016).

Websites Exactly solvable transcendental equations, exactly solvable growth models, optimization, computer assisted research mathematics and its applications priority (CARMA), fast library for number theory (FLINT), etc [108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120] (1999–2017).

Posters the main features of relevance to mathematics [121] (1996), physics and engineering with a contribution to Euler’s tercentenary celebration [122] (2007).

Computational libraries, algorithms, programs (some as open source codes in FORTRAN e.g. wapr.f and matlab wapr.m) with either high or unlimited accuracy (arprec, arblib, lamW) [123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142] (1973–2018).

Approximate, closed formulae for the Lambert function (incorporating the asymptotic behaviors of the Lambert function), e.g. a global approximate formula (a single expression with five adjusted parameters) as a rational function with very good accuracy, or alternatively, a highly accurate approximation using the Padé rational polynomials for the Lambert function [143, 144, 145, 146] (1998–2017).

Articles by Wright Linear and nonlinear differencedifferential equations, solutions of transcendental equations, etc. [147, 148, 149] (1949–1059).

Articles by Siewert et al. Kepler’s problem, Riemann’s problem, critical conditions, the exact solutions of transcendental equations in mathematics and physics, etc. [150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164] (1972–1981).

Articles by Corless et al. Lambert’s W function in Maple, the exact solutions of transcendental equations in mathematics and physics, delayed differential equations, etc. [165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177] (1993–2012).

Articles by Scott et al. Molecular physics (exchange forces for \(\mathrm{H}^+_2\)), general relativity, quantum mechanics, etc. [178, 179, 180, 181, 182, 183, 184] (1993–2012).

Applications in mathematics Solutions to Riemann’s problems for transcendental equations, Siewert–Burniston’s method and its generalization for determining zeros of analytic functions, generalized Gaussian noise model, stiff differential equations, infinite exponentials, series of exponential equations, etc. [185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213] (1952–2018).

Applications to systems with delayed dynamics Stability of delayed systems with repeated poles, delayed fractionalorder dynamic systems, communication networks, multiple delays in synchronization phenomena, bifurcation analysis, characteristic roots of timedelay systems, eigenvalue assignment for control in timedelay systems, timedelayed response of smart material actuator under alternating electric potential, etc. [214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225] (2002–2015).

Applications in physics Corrections in counting detectors, atomic physics (helium eigenfunctions), molecular physics, blackbody radiation, quantum statistics, nonideal diodes in solidstate physics, electromagnetism, acceleratorbased physics (particle storage rings), plasma physics, transport physics (the Fokker–Planck equation), laser physics, thermoelectrics, pair (positron–electron) creation in strong fields, scattering physics, nuclear magnetic resonance (NMR) physics, algorithmic aspects of the Lambert function for problems in physics, a quantummechanical Schrödinger eigenproblem with a potential in the form of the Lambert W function having the exact solution via the confluent hypergeometric function (this potential is of short range and it supports a finite number of bound states), motions of projectiles in media with resistance forces, etc. [226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251] (1980–2016).

Applications in astrophysics Solar winds, solar cells, parametrization of solar photovoltaic system, etc. [252, 253, 254, 255, 256, 257, 258, 259] (2004–2016).

Applications in chemistry Michaelis–Menten enzyme kinetics, NMR for biochemistry, etc. [260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276] (1997–2017).

Applications in biomedicine Epidemics, periodic breathing in chronic heart failure, dark adaptation and the retinoid cycle of vision, infection dynamics, associations/dissociation rate constants of interacting biomolecules, statistical analysis and spatial interpolation in functional magnetic resonance imaging, acidity in solid tumor growth and invasion, a glucoseinsulin dynamic system, blood oxygenation level dependent (BOLD) signals from brain temperature maps, survival of irradiated cells, etc. [277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288] (2000–2015).

Applications in ecology and evolution Euler–Lotka equation, Lotka–Volterra equation, etc. [80, 103] (2009, 2016).

Applications in hydraulics (fluid dynamics) Flow friction, full bore pipe flow within the Colebrook–White equation, etc. [289, 290, 291, 292, 293] (2007–2018).

Applications in energetics and agriculture Moisture content in transformer oil [294] (2013).

Applications in economy Economic order quality: [295] (2012).

Applications in sociology Spread of social phenomena (behaviors, ideas, products), explosive contagion model [296] (2016).

Use of the Lambert function in education Complementing elementary functions by the Lambert function, the Lambert function in the introduction to intermediate physics, the utility of the Lambert function in chemical kinetics, undergraduate theoretical physics eduction, Wien’s displacement law, quantum square well, hanging chain and the gravitational force, etc. [297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 311, 312, 313] (2002–2018).
1.3 Applications using trinomial roots
Trinomial roots attracted a wide interest of researchers over a period longer than 250 years with many interesting and important applications [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52]. In an application of the presently obtained formulae, we will give an example dealing with trinomial roots encountered in radiobiological models for radiotherapy. This illustration concerns cell survival after irradiation for which the measured data from synchronous cell populations ought to be corrected for genome multiplicity [314, 315] prior to appropriate comparisons with the predictions of radiobiological models. Specifically, regarding all but the \(\mathrm{G_1}\) phase cell populations, the corrections of the measured colony surviving fractions F(D) at each dose D need to be made for replications of deoxyribonucleic acid (DNA) molecules, that are the principal radiation target. Such a type of corrections yields a fractional trinomial equation with the sought single cell surviving fraction S(D) raised to power n where \(1\le n\le 2.\) The resulting trinomial roots S(D), amenable to proper comparisons with radiobiological models, are given by a concise analytical formula as the confluent Fox–Wright function \({}_1\Psi _1.\) The results are numerically illustrated on synthesized cell surviving fractions highlighting the competitive roles of genome multiplicity and radiation damage repair as the two components of shoulders in dose–response curves. Our analytical solutions for trinomial roots can also be applied to many other problems, including those with integer powers encountered in e.g. spatiallydependent cell surviving fractions that need to be reconstructed from the measured positron emission densities in imageguided radiotherapy [316].
2 The complete Bell polynomials
3 The cyclic indicator polynomials
4 The partial Bell polynomials
5 Derivatives of any analytical function raised to an arbitrary power
6 An arbitrary power of a MacLaurin series of any function
7 The Lambert series solution for all the roots of trinomial equations
8 All the trinomial roots in terms of the Bell polynomials
9 From multi to univariate polynomials for trinomial roots
10 All the trinomial roots by a series in terms of the Pochhammer symbols
11 Arbitrary real or complexvalued powers of trinomial roots
12 Logarithmic function of trinomial roots
13 Trinomial roots in terms of the confluent Fox–Wright function
14 Convergence radius of the series for trinomial roots
15 An illustration in radiobiology for radiotherapy
As an illustration, we presently carried out the computations on the given synthesized cell survival fractions for two different input data that are either S(D) or F(D) in the cases (i) and (ii), respectively.
Conversely, in the case (ii), which is an inverse or reconstruction problem, the input data are the colony cell surviving fractions F(D). Here, the task is to extract or retrieve the output single cell surviving fraction S(D) from F(D). These reconstructed S(D) data, as the roots of the trinomial equation (15.1), have been computed from the solution (15.3). In the case (ii), the most interesting choice corresponds to F(D) which assumes that no part of DNA has been replicated. With such an input, the output S(D) from (15.1) takes into account DNA replications for \(n>1.\) The input cell colony survival \(F(D)=2S(D)S^n(D)\) is sampled with the linearquadratic (LQ) single cell survival \(S(D)=\exp {(\alpha D\beta D^2)}.\) The ensuing data F(D) for \(n=2\) are shown by the top curve in Fig. 1b. We see that a departure from the straight line, which stems from the term \(\exp {(\alpha D)},\) appears as a prominent shoulder built from two components or mechanisms. One component is cell repair which is described in the LQ model by the Gaussian \(\exp {(\beta D^2)}.\) The other component is DNA replication (or genome multiplicity). Next, starting from the sampled input data F(D) for a fixed n, we reconstruct the output data S(D). As stated, this is done by using (15.3) to compute the roots \(S(D)=[F(D)/2]{}_1\Psi _1([1,n];[2,n1];F^{n1}(D)/2^n)\) of the trinomial equation \(S^n(D)2S(D)+F(D)=0\) from (15.1). The resulting single cell survival data S(D) for \(n=2\) are displayed by the bottom curve in Fig. 1b. This latter curve for S(D) has a reduced shoulder relative to the top curve for F(D). The reason is that S(D) has no contribution from the component due to DNA replications.
Similarly to the case (ii), in measurements with synchronized cell populations, the colony surviving fractions F(D), as the input data to (15.1), contain the contributions from DNA replications and repair. In the output, the single cell surviving fraction S(D) from (15.3) is void of DNA replications. Here, the trinomial equation (15.1) acts as if it were a kind of a “deconvolution” in the sense of removing the unwanted information. The unwanted information is DNA replication which is present in experimental data F(D), but absent from radiobiological models. The desired information S(D), with no replication in any part of DNA, being hidden in F(D), is now unfolded by rooting the trinomial equation (15.1) whose roots are given by (15.3). The ensuing single cell surviving fractions S(D), as the experimental data with no contribution from genome multiplicity, can be used to make the appropriate comparisons with the conventional radiobiological models that, from the onset, ignore DNA replications. In a separate publication, we shall thoroughly investigate this type of radiobiologically important applications, using the measured cell colony surviving fractions from e.g. Refs. [318, 319, 320, 321, 322, 323, 324].
16 Discussion and conclusions
The wellknown theorem by Abel proves that no algebraic solution for the roots of a general nth degree polynomial exists for \(n>4.\) Even in the important case of the simpler, nth degree trinomials, it is not possible to obtain the algebraic roots. An algebraic solution is the exact formula due to a finite number of steps. Of course, numerical computations can give highly accurate values of the roots e.g. by diagonalizing the equivalent Hessenberg or companion matrix which, due to its extreme sparseness, can be of a very high dimension [332, 333, 334]. Nevertheless, it is of interest to find out whether the exact zeros of the nth degree polynomials can be obtained analytically through e.g. an infinite number of steps, as originally suggested by Girard [335]. Such solutions are said to be nonalgebraic and they can occasionally be expressed by certain special functions e.g. transcendental functions, and the like. They can be viewed as certain series or products that involve infinitely many steps. For example, following Girard’s idea [335], it was Lambert [1] who found a series solution of the nth degree trinomial equation \(x^nx+q=0,\) where q is the free, constant term. Subsequently, Euler [3] symmetrized the latter equation as \(x^\alpha x^\beta =(\alpha \beta )v x^{\alpha +\beta }\), where \(\{\alpha ,\beta ,v\}\) are some fixed constants (none of which is necessarily an integer). He solved this equation for the roots x giving a formula as a series (the Euler series). Later, several proofs of Euler’s formula using various methods have been published by a number of authors, including Refs. [11, 20, 33, 34, 35, 36, 45, 47, 50].
The next step regarding the trinomial roots based upon the Euler formula would be to carry out an explicit summation of the Euler’s series. The reason for having such an explicit summation (preferably in a form of one of the known special functions), is in the possibility of exploiting the established features of the identified special function. For example, of particular importance are the asymptotic behaviors of the known special functions at both small and large values of its independent variable. These asymptotes are very useful for analyzing the critical behaviors of the studied system at the two extreme conditions or situations.

First, we carry out the proof of the Euler’s formula by deriving the general expansion coefficient of the Euler series in terms of the complete multivariate Bell polynomial \(B_n\) (also called the exponential polynomial).

Second, the multivariate Bell polynomial is reduced to a much simpler univariate polynomial in terms of either the Pochhammer symbol or binomial coefficients.

Third, the Pochhammer simplification enables the identification of the transformed series as a special function called the confluent Fox–Wright function \({}_1\Psi _1.\)

Fourth, another confluent Fox–Wright function \({}_1\Psi _1\) is also found for an arbitrary power (any real or complex constant) of the derived trinomial roots.

Fifth, the logarithm of the trinomial root is expressed through a single series with the expansion coefficients in the form of either the Pochhammer symbols or the binomial coefficients.
An illustration is given using synthesized data for survival of irradiated cells. The simulations are reminiscent of the corresponding measured colony surviving fractions for Chinese hamster synchronized cell populations exposed to 250 kVp Xrays (see the survival curve in e.g. Fig. 9 from Ref. [318] for 10.4 h after incubation). Shown in the present Fig. 1 are the single cell surviving fractions S(D) and the cell colony surviving fractions F(D) as a function of radiation instantaneous dose D. Panels (a) and (b) on Fig. 1 are on the direct and inverse problems, respectively. Both panels in this figure deal with a relationship between S(D) and F(D). This is a trinomial relationship, \(S^n(D) 2S(D)+F(D)=0 \, (1\le n\le 2),\) from correcting S(D) for the missing genome multiplicity n (DNA replications) [314, 315]. In Fig. 1a, the input and output data are S(D) and F(D), respectively. Conversely, in Fig. 1b, the input and output data are F(D) and S(D), respectively. The input bottom curve in Fig. 1a is a purely exponential survival S(D) as a straight line on a semilogarithmic scale. When this S(D) is corrected for genome multiplicity with \(n=1.5\) and \(n=2,\) the middle and the top curves are obtained in Fig. 1a for the output data \(F(D)=2S(D)S^n(D),\) respectively. Here, a clearly delineated shoulder appears in the top curve of F(D) for \(n=2.\) In Fig. 1b, the input data F(D), given by the top curve, are the linearquadratic colony surviving fractions corrected for genome multiplicity with \(n=2.\) Here, a pronounced shoulder is built from two components: DNA replication and radiation damage repair. When such digitized F(D) data are inserted into the trinomial equation \(S^n(D) 2S(D)+F(D)=0,\) its twovalued roots S(D) are obtained for \(n=2\) (computation is carried out from the present series solution for the single cell survival S(D) in terms of the Fox–Wright function \({}_1\Psi _1\)). The smaller of the two roots, as the physical solution, is shown by the bottom curve in Fig. 1b. Therein, because S(D) is void of the contribution from DNA replications, a diminished shoulder (due to repair alone) is seen in the bottom curve for S(D). This is clear by reference to the top curve in Fig. 1b for F(D) whose shoulder contains both DNA replication and cell repair. Such observations are anticipated to be a further motivation for additional explorations of surviving fractions corrected for genome multiplicity. These corrections are necessary for cell populations in all the phases of the cell reproduction cycle, except the \(\mathrm{G}_1\)phase cell population.
Notes
Acknowledgements
This work is supported by the research grants from Radiumhemmet at the Karolinska University Hospital and the City Council of Stockholm (FoUU) to which the author is grateful.
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