Confluent hypergeometric function
The confluent hypergeometric function is a class of special functions that arise as solutions to Kummer's confluent hypergeometric differential equation, a second-order linear ordinary differential equation of the form z w'' + (b - z) w' - a w = 0, where a and b are parameters, featuring a regular singularity at z = 0 with indices 0 and $1 - b, and an irregular singularity at infinity of rank one.[1] This equation represents a limiting case of the Gauss hypergeometric differential equation obtained by coalescing two regular singularities at finite points into one.[1] The two linearly independent solutions are the Kummer function of the first kind, denoted M(a, b, z) and defined by the power series M(a, b, z) = \sum_{s=0}^{\infty} \frac{(a)_s}{(b)_s} \frac{z^s}{s!} (convergent for all finite complex z, entire in z and a, and meromorphic in b provided b is not a nonpositive integer), and the Tricomi function of the second kind, denoted U(a, b, z), which exhibits the asymptotic behavior U(a, b, z) \sim z^{-a} as |z| \to \infty in |\arg z| < \frac{3\pi}{2}.[1] Introduced by Ernst Kummer in 1837 as part of his investigations into hypergeometric series, the confluent hypergeometric function—often simply called Kummer's function—has become a cornerstone of mathematical analysis due to its connections to other special functions, including the error function, Bessel functions, parabolic cylinder functions, and Laguerre polynomials (via the limiting case L_n^{(\alpha)}(z) = \frac{(1 + \alpha)_n}{n!} \, _1F_1(-n; \alpha + 1; z), where _1F_1 denotes the confluent hypergeometric function in Pochhammer notation).[2][3] Key properties include Kummer's transformation M(a, b, z) = e^z M(b - a, b, -z), analytic continuation formulas, and integral representations such as M(a, b, z) = \frac{\Gamma(b)}{\Gamma(a) \Gamma(b - a)} \int_0^1 e^{z t} t^{a-1} (1 - t)^{b - a - 1} \, dt for \Re b > \Re a > 0.[1] These functions are implemented in major mathematical software libraries and serve as building blocks for solving linear differential equations with variable coefficients.[4] The confluent hypergeometric functions find extensive applications in mathematical physics and applied mathematics, particularly in quantum mechanics where they provide exact solutions to the radial Schrödinger equation for the hydrogen atom (expressed in terms of associated Laguerre polynomials derived from M) and other solvable potentials like the Morse and Pöschl-Teller potentials.[5] They also appear in the exact solutions of the wave equation in paraboloidal coordinates, Coulomb scattering problems, and many-body dynamics in statistical mechanics.[5] In group theory, the functions are linked to irreducible representations of third-order triangular matrix groups.[6] Asymptotic expansions and numerical methods for their computation, such as those developed for large parameters, further underscore their utility in engineering and computational science.[7]Introduction and Definition
Historical Background
The confluent hypergeometric function arises as a limiting case of the Gaussian hypergeometric function _2F_1(a,b;c;z) in which one upper parameter tends to infinity while the argument is scaled accordingly, yielding M(a,c,z)=\lim_{b\to\infty}{}_2F_1(a,b;c;z/b).[1] This degeneration process coalesces two regular singularities of the associated hypergeometric differential equation into a single irregular singularity at infinity, simplifying the structure while preserving key analytic properties.[1] Introduced by Ernst Kummer in his foundational 1836 study of hypergeometric series, the function emerged from efforts to solve second-order linear differential equations through power series methods. Kummer's work focused on systematic expansions and convergence properties, establishing the confluent form as a natural extension of earlier hypergeometric investigations by Gauss and others.[8] Subsequent development by Henri Poincaré in 1885 further refined its role within the broader theory of linear differential equations, emphasizing transformations and singularity analysis.[9] The primary motivations for studying the confluent hypergeometric function stemmed from its appearance in confluent versions of hypergeometric differential equations encountered in 19th-century problems of mathematical physics.[10] A key milestone in its compilation and dissemination came with Harry Bateman's 1953 handbook, which systematically gathered properties, identities, and integral representations from scattered literature, facilitating broader adoption in applied mathematics.[11]Basic Definition
The confluent hypergeometric function of the first kind, known as Kummer's function and denoted by M(a, b, z) or {}_1F_1(a; b; z), serves as a fundamental special function in mathematical analysis. A second linearly independent solution to the associated differential equation is provided by Tricomi's function U(a, b, z). These functions are parameterized by complex numbers a, b, and z, with b \notin \{0, -1, -2, \dots\} to ensure well-defined behavior.[1][1][12] Central to their definitions is the Pochhammer symbol, or rising factorial, given by (a)_n = a(a+1)\cdots(a+n-1) for nonnegative integer n, with the convention (a)_0 = 1. This symbol facilitates the expression of the functions' properties and expansions.[1] For fixed a and b (satisfying the condition on b), M(a, b, z) is holomorphic throughout the complex z-plane. Analytic continuation extends its definition to all complex a, b, and z under the same restriction on b. Normalization properties include M(0, b, z) = 1 and M(a, b, 0) = 1 for admissible parameters.[12][12][1][1]Differential Equation
Kummer's Equation
The confluent hypergeometric equation, known as Kummer's equation, is given by z y'' + (b - z) y' - a y = 0, where a and b are complex parameters, with b \notin \{0, -1, -2, \dots \}. This second-order linear ordinary differential equation has two linearly independent solutions: the confluent hypergeometric function of the first kind M(a, b, z) and of the second kind U(a, b, z).[1] Kummer's equation arises as the confluent limit of Gauss's hypergeometric differential equation z(1-z) y'' + [c - (a+b+1)z] y' - ab y = 0. To obtain it, substitute z \to z/b into the hypergeometric equation, let b \to \infty, and set the hypergeometric parameter c = b; this process causes two regular singular points to coalesce into one regular singular point and one irregular singular point.[1][13] The equation possesses a regular singular point at z = 0 and an irregular singular point at z = \infty of rank one. Around the regular singular point z = 0, solutions can be constructed using the Frobenius method, which assumes a series expansion of the form y = z^\lambda \sum_{n=0}^\infty c_n z^n. Substituting into the equation yields the indicial equation \lambda(\lambda - 1 + b) = 0, with roots \lambda = 0 and \lambda = 1 - b; the solution corresponding to \lambda = 0 (assuming b is not an integer) is proportional to M(a, b, z).[1][13] The solutions M(a, b, z) and U(a, b, z) are linearly independent, as evidenced by their Wronskian W(M(a, b, z), U(a, b, z)) = -\frac{z^{-b} e^z}{\Gamma(a)}. This Wronskian confirms the independence where the functions are defined and appropriate branch choices are made.[1]Related Differential Equations
The Whittaker equation provides an alternative canonical form for the differential equation satisfied by confluent hypergeometric functions, obtained via a change of dependent and independent variables from the standard Kummer equation.[14] Specifically, substituting w(z) = e^{-z/2} z^{\mu + 1/2} M(a, b, z) with \kappa = b/2 - a and \mu = (b-1)/2 transforms Kummer's equation into Whittaker's equation: \frac{d^2 w}{dz^2} + \left( -\frac{1}{4} + \frac{\kappa}{z} + \frac{\frac{1}{4} - \mu^2}{z^2} \right) w = 0. The linearly independent solutions are the Whittaker functions M_{\kappa, \mu}(z) and W_{\kappa, \mu}(z), defined as M_{\kappa, \mu}(z) = e^{-z/2} z^{\mu + 1/2} \, _1F_1\left( \frac{1}{2} + \mu - \kappa; 1 + 2\mu; z \right) and W_{\kappa, \mu}(z) = e^{-z/2} z^{\mu + 1/2} U\left( \frac{1}{2} + \mu - \kappa; 1 + 2\mu; z \right), where _1F_1 and U denote the Kummer confluent hypergeometric function of the first and second kind, respectively.[14] This substitution highlights the regular singularity at z=0 (with indices \frac{1}{2} \pm \mu) and the irregular singularity at infinity, preserving the structure of the original equation while facilitating applications in quantum mechanics and other fields.[14] Inversely, the Kummer function relates to the Whittaker function via _1F_1(a; b; z) = e^{z/2} z^{-b/2} M_{b/2 - a, (b-1)/2}(z), demonstrating that solutions to Whittaker's equation encompass the confluent hypergeometric functions.[14] The parabolic cylinder differential equation represents a special limiting case of the confluent hypergeometric equation, arising when the parameters align to produce a quadratic potential term.[15] The standard form is \frac{d^2 u}{dz^2} - \left( \frac{1}{4} z^2 + a \right) u = 0, with fundamental solutions U(a, z) and V(a, z), the parabolic cylinder functions.[16] This equation connects directly to the confluent hypergeometric form through parameter substitutions, such as expressing U(a, z) in terms of U\left( \frac{a+1}{2}, \frac{3}{2}, \frac{z^2}{2} \right) or M\left( \frac{a}{2} + \frac{1}{4}, \frac{1}{2}, \frac{z^2}{2} \right).[15] For instance, M\left( \frac{1}{2} a + \frac{1}{4}, \frac{1}{2}, \frac{1}{2} z^2 \right) = \frac{2^{\frac{1}{2} a - \frac{3}{4}} \Gamma\left( \frac{1}{2} a + \frac{3}{4} \right) e^{\frac{1}{4} z^2} }{\sqrt{\pi}} \left[ U(a, z) + U(a, -z) \right], illustrating how confluent hypergeometric solutions reduce to parabolic cylinder functions under specific conditions, particularly useful in solving Weber's equation or quantum harmonic oscillator problems.[15] Confluent hypergeometric functions also relate to Laplace's equation through transformed variables, notably in parabolic coordinates where separation of variables yields the parabolic cylinder equation as the radial (or angular) component.[17] In this context, substituting parabolic coordinates (\sigma, \tau) into \nabla^2 \phi = 0 produces ordinary differential equations solvable by confluent hypergeometric functions via the parabolic cylinder special case, enabling exact solutions for axisymmetric potentials.[17]Representations
Power Series Expansion
The confluent hypergeometric function of the first kind, denoted M(a, b, z), admits a power series expansion around z = 0: M(a, b, z) = \sum_{n=0}^{\infty} \frac{(a)_n}{(b)_n} \frac{z^n}{n!}, where (a)_n and (b)_n are Pochhammer symbols defined as rising factorials, with (a)_0 = 1 and (a)_{n+1} = (a)_n (a + n) for n \geq 0. This representation holds provided b is not a non-positive integer, in which case the function may reduce to a polynomial or require analytic continuation.[1] The series converges for all finite complex values of z, establishing M(a, b, z) as an entire function in z. This infinite radius of convergence follows from the ratio test applied to the coefficients, where the limit of the absolute value of the ratio of consecutive terms is zero as n \to \infty.[1] For practical computation, the series is truncated when terms become sufficiently small, with rapid convergence for small |z|. This power series solution arises from applying the Frobenius method to Kummer's differential equation, z \frac{d^2 w}{dz^2} + (b - z) \frac{d w}{dz} - a w = 0, which has a regular singular point at z = 0. Assume a series solution of the form w(z) = z^r \sum_{n=0}^{\infty} c_n z^n with c_0 \neq 0. Substituting into the equation yields the indicial equation r(r-1) + b r = 0, with roots r = 0 and r = 1 - b. For the root r = 0, the recurrence relation for the coefficients is (n+1)(b + n) c_{n+1} = (a + n) c_n, \quad n \geq 0, which, starting from c_0 = 1, gives c_n = \frac{(a)_n}{(b)_n n!}. This directly produces the series for M(a, b, z), normalized such that M(a, b, 0) = 1.[18] The second solution corresponding to r = 1 - b involves a logarithmic term unless b is an integer, but the power series form focuses on the regular solution at the origin.[1]Integral Representations
The confluent hypergeometric function M(a, b, z) possesses several integral representations that facilitate its analytic continuation beyond the regions where the power series converges, particularly useful when parameter values lead to divergences in other forms. These include definite integrals over finite intervals, Mellin-Barnes contour integrals along vertical lines in the complex plane, and loop-type contour integrals such as the Pochhammer contour, each valid under specific conditions on the real parts of the parameters and the argument z.[19] A fundamental definite integral representation is given by M(a, b, z) = \frac{\Gamma(b)}{\Gamma(a) \Gamma(b - a)} \int_0^1 e^{z t} t^{a-1} (1 - t)^{b - a - 1} \, dt, valid for \Re b > \Re a > 0. This form expresses M(a, b, z) as a weighted average of exponentials over the unit interval, leveraging the beta function for normalization, and converges due to the polynomial growth of the integrand bounded by the exponential. It is particularly effective for numerical computation when z is not too large in magnitude. The Mellin-Barnes integral provides a contour representation suitable for asymptotic analysis and continuation: M(a, b, -z) = \frac{1}{2 \pi i \Gamma(a)} \int_{-i \infty}^{i \infty} \frac{\Gamma(a + t) \Gamma(-t)}{\Gamma(b + t)} z^t \, dt, where the vertical contour separates the poles of \Gamma(a + t) (at t = -a - n, n = 0, 1, 2, \dots) from those of \Gamma(-t) (at t = 0, [1](/page/1), 2, \dots), requiring a \neq 0, -1, -2, \dots and |\arg z| < \pi/2 for convergence. This Barnes-type integral arises from the Mellin transform of the power series expansion and allows deformation of the contour to capture residues corresponding to the series terms. For cases involving multi-valued behavior or when \Re b > \Re a > 0 does not hold, loop integrals over Pochhammer contours extend the representation. One such form is M(a, b, z) = \frac{\Gamma(1 + a - b)}{2 \pi i \Gamma(a)} \int_0^{(1+)} e^{z t} t^{a-1} (t - 1)^{b - a - 1} \, dt, where the contour starts at t = 0, proceeds to just above the real axis to encircle t = 1 positively, and returns to t = 0 just below the real axis, valid for b - a \neq 1, 2, 3, \dots and \Re a > 0. A related double-loop Pochhammer contour representation is M(a, b, z) = e^{-b \pi i} \Gamma(1 - a) \Gamma(1 + a - b) \frac{1}{4 \pi^2} \int_\alpha^{(0+, 1+, 0-, 1-)} e^{z t} t^{a-1} (1 - t)^{b - a - 1} \, dt, with the contour, denoted ∫_α^(0⁺,1⁺,0⁻,1⁻), starting at a point α ∈ (0,1), encircling t=0 positively then t=1 positively, followed by encircling t=0 negatively then t=1 negatively, before returning to α, applicable when a, b - a \neq 1, 2, 3, \dots. These contours avoid branch cuts and enable evaluation in regions where real-line integrals diverge, provided the exponential factor ensures convergence at infinity.[19]Asymptotic Behavior
Large Argument Asymptotics
The asymptotic behavior of the confluent hypergeometric functions for large |z| is crucial for understanding their growth or decay in various sectors of the complex plane. The Kummer function M(a, b, z) exhibits exponential growth in the principal sector, while the Tricomi function U(a, b, z) provides a recessive solution that decays algebraically. These expansions are derived using the method of steepest descent applied to the integral representations of the functions, which allows for systematic determination of the leading contributions from saddle points or endpoints.[20] For M(a, b, z), in the sector where |\arg z| < \pi/2, the dominant asymptotic behavior is given by the exponentially growing term: M(a, b, z) \sim \frac{\Gamma(b)}{\Gamma(a)} \, e^{z} z^{a - b} \left(1 + O\left(\frac{1}{z}\right)\right), assuming \operatorname{Re} b > \operatorname{Re} a > 0 and non-integer differences to avoid logarithmic terms. This leading approximation captures the rapid increase along the positive real axis, with higher-order terms obtainable from the full Poincaré series expansion.[20][21] In contrast, U(a, b, z) is the solution that remains bounded or decays for large |z| in a wider sector, |\arg z| < 3\pi/2. Its leading asymptotic form is U(a, b, z) \sim z^{-a} \left(1 + O\left(\frac{1}{z}\right)\right), valid under similar parameter conditions. This algebraic decay makes U(a, b, z) particularly useful for applications requiring recessive behavior at infinity.[20] A complete asymptotic series for U(a, b, z) in inverse powers of z is U(a, b, z) \sim z^{-a} \sum_{n=0}^{\infty} (-1)^{n} \frac{(a)_{n} (a - b + 1)_{n}}{n! \, z^{n}}, which provides corrections beyond the leading term and is uniformly valid in the specified sector, excluding Stokes lines where subdominant contributions may emerge. This series arises directly from the steepest descent analysis of the integral form and terminates for certain integer parameters.[20]Stokes Phenomenon
The Stokes phenomenon in the confluent hypergeometric function M(a, b, z) describes the discontinuous change in the coefficients of subdominant terms within its asymptotic expansions as certain rays, called Stokes lines, are crossed in the complex z-plane. These lines are defined as the loci where the imaginary part of the exponent in the dominant term vanishes, specifically at \arg z = (2k+1)\pi for integer k, such that crossing them activates exponentially small subdominant contributions that were negligible in adjacent sectors. This switching ensures the analytic continuation of the function across the plane but introduces jumps in the formal asymptotic series, requiring careful sectorial validity analysis.[22][23] Across anti-Stokes lines, which separate regions of dominance, the multiplier for the subdominant part of M(a, b, z) undergoes a transition captured by Stokes multipliers, often approximated by factors like \cos(\pi a) in leading order for real parameters. More precisely, the full multiplier incorporates a smooth variation near the Stokes line itself, modeled by the complementary error function \operatorname{erfc} to describe the rapid onset of the subdominant exponential: for instance, the effective multiplier S near \arg z = \pi takes the form S \approx \frac{1}{2} + \frac{1}{2} \operatorname{erf}(\alpha \sqrt{\pi}), where \alpha depends on the imaginary component of the phase function relative to its real part. This error function smoothing resolves the apparent discontinuity, providing a uniform approximation in transitional sectors.[22][23] In regions transitional to Stokes lines, the asymptotic behavior of M(a, b, z) connects directly to parabolic cylinder functions, such as D_\nu(z), through parameter transformations like M(a, b, z) = e^{-z/2} z^{(b-1)/2} D_{b-2a-1}(\sqrt{2z}) for appropriate scalings, where the parabolic cylinder exhibits analogous subdominant switching. Uniform expansions incorporating these functions, often via hyperasymptotics, bridge the dominant and subdominant regimes, with remainders bounded exponentially better than standard truncations—for example, error reductions from O(10^{-3}) to O(10^{-5}) or smaller in test cases near \arg z = \pi.[24][25][23] Numerically, accounting for the Stokes phenomenon is essential for evaluating M(a, b, z) in complex domains, as ignoring the subdominant activation across lines can yield errors exceeding 10% even at moderate |z| (e.g., |z| = 20), particularly in applications requiring high precision like quantum mechanics or heat conduction. Hyperasymptotic re-expansions, which resum optimal truncations with error function transitions, enable stable computations by exponentially improving accuracy—achieving factors of e^{-c \sqrt{|z|}} in remainder estimates—thus facilitating reliable software implementations for arbitrary arguments.[25][22][23]Functional Relations
Contiguous Relations
The contiguous relations for the confluent hypergeometric function of the first kind M(a, b, z) are linear recurrence relations that connect it to other instances of the function whose parameters differ by integers, specifically \pm 1. These relations arise naturally from the structure of the function and are crucial for analytical manipulations and numerical algorithms that compute M(a, b, z) via stepwise parameter adjustments, ensuring stability in regions where direct series summation may fail. There are six fundamental contiguous functions associated with M(a, b, z): M(a+1, b, z), M(a-1, b, z), M(a, b+1, z), and M(a, b-1, z). The relations typically involve linear combinations of three such functions with coefficients that are polynomials in the parameters a, b and the argument z.[26] The complete set of six basic contiguous relations is given by: \begin{align} &(b - a) M(a-1, b, z) + (2a - b + z) M(a, b, z) - a M(a+1, b, z) = 0, \tag{13.3.1} \\ &b(b-1) M(a, b-1, z) + b(1 - b - z) M(a, b, z) + z(b - a) M(a, b+1, z) = 0, \tag{13.3.2} \\ &(a - b + 1) M(a, b, z) - a M(a+1, b, z) + (b - 1) M(a, b-1, z) = 0, \tag{13.3.3} \\ &b M(a, b, z) - b M(a-1, b, z) - z M(a, b+1, z) = 0, \tag{13.3.4} \\ &b(a + z) M(a, b, z) + z(a - b) M(a, b+1, z) - a b M(a+1, b, z) = 0, \tag{13.3.5} \\ &(a - 1 + z) M(a, b, z) + (b - a) M(a-1, b, z) + (1 - b) M(a, b-1, z) = 0. \tag{13.3.6} \end{align} These identities hold for parameters where the functions are defined, typically assuming b \neq 0, -1, -2, \dots to avoid poles.[26] These relations can be derived by substituting the power series expansion M(a, b, z) = \sum_{n=0}^{\infty} \frac{(a)_n}{(b)_n} \frac{z^n}{n!} into each equation and verifying coefficient-wise equality, or alternatively by applying differentiation with respect to the parameters a or b to Kummer's differential equation z y'' + (b - z) y' - a y = 0 and combining the resulting equations. For instance, shifting b to b+1 in the differential equation and eliminating higher derivatives using the original equation yields relations like (13.3.4).[26][1] Contiguous relations also interconnect with derivative formulas, such as the first-order relation \frac{d}{dz} M(a, b, z) = \frac{a}{b} M(a+1, b+1, z), allowing expression of derivatives in terms of diagonally contiguous functions; combining this with parameter-shift relations like (13.3.4) facilitates expressions such as z \frac{d}{dz} M(a, b, z) = b \left[ M(a, b, z) - M(a, b+1, z) \right]. Higher-order derivatives follow analogously: \frac{d^n}{dz^n} M(a, b, z) = \frac{(a)_n}{(b)_n} M(a+n, b+n, z).[26] In practice, these relations enable efficient recurrence-based computations of M(a, b, z), particularly for large or negative parameters where forward or backward recurrences avoid numerical instability, as detailed in algorithms for special function evaluation.[26]Kummer's Transformation
Kummer's transformation establishes a fundamental symmetry for the confluent hypergeometric function of the first kind, relating its value at argument z to that at -z. The identity is given by M(a, b, z) = e^{z} \, M(b - a, b, -z), valid for all complex a, b \neq 0, -1, -2, \dots, and z, where M denotes the Kummer function of the first kind. This relation was introduced by Kummer in his foundational work on hypergeometric series. The transformation can be proved using the power series representation of M(a, b, z), M(a, b, z) = \sum_{n=0}^{\infty} \frac{(a)_n}{(b)_n} \frac{z^n}{n!}, where (\cdot)_n is the Pochhammer symbol. Substituting the series for M(b - a, b, -z) into the right-hand side yields e^{z} \, M(b - a, b, -z) = \left( \sum_{k=0}^{\infty} \frac{z^k}{k!} \right) \left( \sum_{n=0}^{\infty} \frac{(b - a)_n (-z)^n}{(b)_n n!} \right). The coefficient of z^m in the product expansion is \frac{1}{m!} \sum_{k=0}^{m} \binom{m}{k} (-1)^{m-k} \frac{(b - a)_{m-k}}{(b)_{m-k}}, which simplifies to \frac{(a)_m}{(b)_m} via the chu-vandermonde identity for hypergeometric series, matching the series for M(a, b, z). An alternative proof employs the integral representation (for \operatorname{Re} b > \operatorname{Re} a > 0), M(a, b, z) = \frac{\Gamma(b)}{\Gamma(a) \Gamma(b - a)} \int_{0}^{1} e^{z t} t^{a-1} (1 - t)^{b - a - 1} \, dt. Applying the substitution t \to 1 - t transforms the integral to one representing e^z M(b - a, b, -z), confirming the identity. For the confluent hypergeometric function of the second kind U(a, b, z), the transformation follows from the connection formulas linking U to M, yielding U(a, b, z) = z^{1 - b} \, U(a - b + 1, 2 - b, z), valid under the same parameter conditions, with branches chosen appropriately for complex z. This relation arises by applying Kummer's transformation to the expressions for U in terms of M. Generalizations extend the transformation to complex arguments with phase adjustments. For instance, if \arg(-z) = \arg z + \pi, M(a, b, z) = e^{z} (-1)^{a - b} \, M(b - a, b, -z), accounting for the principal branch of the Pochhammer symbols; similar adjustments apply to U. These ensure analytic continuation across branch cuts.Multiplication Theorem
The multiplication theorems for Kummer's confluent hypergeometric functions express M(a, b, xy) and U(a, b, xy) as infinite series involving the functions with unscaled argument x but shifted parameters. These theorems are derived by substituting y → (y − 1)x into the addition theorems of §13.13(i) and §13.13(ii), respectively, yielding expansions valid in specified regions of the complex plane. The resulting formulas facilitate the analysis of the functions under argument scaling, particularly when direct computation of the scaled form is challenging, and are useful in applications requiring parameter transformations or asymptotic approximations.[27] One standard form for the function of the first kind is M(a, b, xy) = e^{(y-1)x} \sum_{n=0}^{\infty} \frac{(b-a)_n [-(y-1)x]^n}{(b)_n n!} M(a, b + n, x), valid for |y − 1| < 1. An alternative expansion, suitable for Re(y) > −1/2, is M(a, b, xy) = y^{-a} \sum_{n=0}^{\infty} \frac{(a)_n (y-1)^n}{n! y^n} M(a + n, b, x). These expressions shift the parameter b or a while keeping the argument fixed at x, allowing recursive or iterative computations. The addition theorems from which they are derived rely on contiguous relations—differential and recurrence relations connecting functions with parameters differing by integers—applied to the power series expansion of M(a, b, z).[27] For the function of the second kind, a corresponding formula is U(a, b, xy) = e^{(y-1)x} \sum_{n=0}^{\infty} \frac{[-(y-1)x]^n}{n!} U(a, b + n, x), valid for |y − 1| < 1. Other variants follow similarly from the remaining addition theorems, with convergence conditions such as Re(y) > −1/2 or |y − 1| < 1 ensuring the series converge. For non-integer scaling factors y, these sum representations are the primary form, though integral representations of M and U can alternatively be scaled directly via variable substitution to derive equivalent expressions. When the scaling y = m is a positive integer, the formulas specialize without alteration, remaining infinite sums unless a or b − a is a non-positive integer, in which case the Pochhammer symbols cause termination, yielding finite sums.[27] A notable special case occurs for m = 2, where the theorems link to representations of the error function erf(z) and complementary error function erfc(z), which are special values of the confluent hypergeometric functions: erf(z) = (2z / √π) M(1/2, 3/2, −z²) and erfc(z) = (e^{−z²} / √π z) U(1/2, 3/2, z²). Applying the multiplication formula with y = 2 to these parameter values yields relations that connect erf(√2 z) or similar scaled arguments to sums involving erf(z) or erfc(z) at z, facilitating computations in probability and heat conduction problems where scaled error functions arise. The derivation of these links follows from substituting the specific parameters into the general sum and using known asymptotic or series identities for the error functions.[1]Connections to Other Functions
Laguerre Polynomials
The associated Laguerre polynomials L_n^{(\alpha)}(z), which generalize the standard Laguerre polynomials when \alpha = 0, are directly expressed in terms of the confluent hypergeometric function of the first kind M(a, b, z). Specifically, for nonnegative integer n and \alpha > -1, L_n^{(\alpha)}(z) = \frac{(\alpha + 1)_n}{n!} \, M(-n, \alpha + 1, z), where (\alpha + 1)_n denotes the Pochhammer symbol (rising factorial). This relation highlights how the confluent hypergeometric function serves as a generating mechanism for these polynomials: when the first parameter a = -n is a negative integer, the power series of M(a, b, z) terminates after n+1 terms, producing a polynomial of degree n. As briefly referenced in the power series expansion section, this termination property arises from the structure of the hypergeometric series. The generating function for the associated Laguerre polynomials further connects them to confluent hypergeometric forms, though it simplifies due to the specific parameters involved. The ordinary generating function is given by \sum_{n=0}^{\infty} L_n^{(\alpha)}(z) t^n = (1 - t)^{-\alpha - 1} \exp\left( \frac{z t}{t - 1} \right), valid for |t| < 1. This expression can be rewritten noting that \exp\left( \frac{z t}{t - 1} \right) = \exp\left( -\frac{z t}{1 - t} \right) and that the exponential function is a special case of the confluent hypergeometric function when the upper and lower parameters coincide, since M(a, a, w) = e^w. These polynomials inherit key properties from the confluent hypergeometric function, including orthogonality on the interval (0, \infty) with respect to the weight function w(x) = e^{-x} x^{\alpha} for \alpha > -1. The orthogonality relation is \int_0^{\infty} e^{-x} x^{\alpha} L_n^{(\alpha)}(x) L_m^{(\alpha)}(x) \, dx = \frac{\Gamma(n + \alpha + 1)}{n!} \delta_{mn}, where \delta_{mn} is the Kronecker delta. This property stems from the terminating hypergeometric series and the integral representations of M(a, b, z), which ensure the polynomials form an orthogonal basis under this weighted inner product. Additional inherited features include contiguous relations and recurrence formulas derived from those of the confluent hypergeometric function, facilitating computations and extensions. In the limit as n \to \infty, the associated Laguerre polynomials recover the full non-terminating confluent hypergeometric function, illustrating the generalization: while finite n yields polynomials via the truncated series of M(-n, \alpha + 1, z), the infinite series expansion of M(a, \alpha + 1, z) for general a extends these to transcendental functions with broader analytic behavior. This limiting process underscores how the confluent hypergeometric function encompasses the polynomial case as a special instance, with applications in approximation theory and spectral methods relying on this interplay.Whittaker and Tricomi Functions
The Whittaker functions provide a standardized parametrization of solutions to the confluent hypergeometric equation, particularly suited for problems with singularities at zero and infinity. These functions, introduced by E. T. Whittaker, are defined for parameters \kappa and \mu with \operatorname{Re} z > 0 and \operatorname{Re} (1 + 2\mu) > 0 to ensure convergence. The regular Whittaker function of the first kind is given by M_{\kappa,\mu}(z) = e^{-z/2} z^{\mu + 1/2} \, {}_1F_1\left(\mu - \kappa + \frac{1}{2}; 1 + 2\mu; z\right), where {}_1F_1 denotes Kummer's confluent hypergeometric function M(a, b, z). This form incorporates the exponential factor e^{-z/2} and power z^{\mu + 1/2} to normalize the behavior near z = 0, where M_{\kappa,\mu}(z) \sim z^{\mu + 1/2}, and facilitates applications in radial equations. The Whittaker function of the second kind, W_{\kappa,\mu}(z), is irregular at z = 0 and asymptotically dominant for large |z|, defined as W_{\kappa,\mu}(z) = e^{-z/2} z^{\mu + 1/2} \, U\left(\mu - \kappa + \frac{1}{2}; 1 + 2\mu; z\right), with U(a, b, z) being Tricomi's confluent hypergeometric function of the second kind. This normalization ensures W_{\kappa,\mu}(z) \sim e^{-z/2} z^\kappa as |z| \to \infty in |\arg z| < \frac{3\pi}{2}, making it the principal branch for asymptotic analysis. Both functions satisfy Whittaker's equation, \frac{d^2 w}{dz^2} + \left( -\frac{1}{4} + \frac{\kappa}{z} + \frac{\frac{1}{4} - \mu^2}{z^2} \right) w = 0, which arises from Kummer's equation via the substitution w = e^{z/2} z^{-\mu - 1/2} W. Tricomi's function U(a, b, z), introduced by F. G. Tricomi, serves as the independent second solution to Kummer's equation alongside M(a, b, z), with the principal branch defined for \arg z \in (-\pi, \pi). It is particularly valuable for its asymptotic decay U(a, b, z) \sim z^{-a} as |z| \to \infty, enabling the construction of W_{\kappa,\mu}(z) for physical problems requiring bounded solutions at infinity. The interconversion between Whittaker functions and the standard confluent forms is direct via the above definitions, allowing expressions like U(a, b, z) = \frac{W_{\frac{1}{2}(b - 2a), \frac{1}{2}(b-1)}(z)}{e^{-z/2} z^{b/2}} for appropriate parameters. This parametrization is advantageous in quantum mechanics, where Whittaker functions solve the radial Schrödinger equation for potentials like the Coulomb or hydrogen atom, providing normalized wave functions with clear separation of angular and radial behaviors.[14]Special Cases
Reduction to Elementary Functions
The confluent hypergeometric function of the first kind, denoted M(a, b, z), simplifies to the elementary exponential function when the parameters satisfy b = a, yielding M(a, a, z) = e^{z}. This identity holds for all complex z and parameters a where the function is defined, providing a direct reduction without additional conditions. Similarly, when a = 0, the function reduces to the constant M(0, b, z) = 1 for b \neq 0, -1, -2, \dots. Further reductions occur in relation to the incomplete gamma function under specific parameter constraints. For instance, if b = a + 1, then M(a, a+1, -z) = a z^{-a} \gamma(a, z), where \gamma(a, z) = \int_0^z t^{a-1} e^{-t} \, dt is the lower incomplete gamma function, valid for \operatorname{Re} a > 0 and \operatorname{Re} z > 0. This form is elementary when a takes values such as positive integers, where \gamma(a, z) expresses in terms of finite sums and exponentials, or half-integers like a = 1/2, relating to the error function via \gamma(1/2, z) = \sqrt{\pi} \, \operatorname{erf}(\sqrt{z}). In the latter case, a related expression is M(1/2, 3/2, -z^2) = \frac{\sqrt{\pi}}{2z} \operatorname{erf}(z), connecting directly to the error function \operatorname{erf}(z) = \frac{2}{\sqrt{\pi}} \int_0^z e^{-t^2} \, dt. More generally, when b - a is a positive integer, say b = a + m with m = 1, 2, \dots, the function M(a, b, z) can be expressed using the incomplete gamma function through contiguous relations or direct summation, assuming \operatorname{Re} a > 0.[15] For negative integer a = -n with n = 0, 1, 2, \dots, M(-n, b, z) terminates as a polynomial of degree n, which is elementary; this is the generalized Laguerre polynomial when normalized appropriately. The second kind function U(a, b, z) also reduces elementarily in parallel cases, such as U(a, a+1, z) = z^{-a} and connections to the exponential integral E_a(z) = \int_1^\infty e^{-z t} t^{-a} \, dt for \operatorname{Re} z > 0.Specific Parameter Values
When the parameter a = 0, the confluent hypergeometric function simplifies to M(0, b, z) = 1 for all b \notin \{0, -1, -2, \dots\} and all finite z \in \mathbb{C}. Similarly, M(a, b, 0) = 1 holds for all finite a and b \notin \{0, -1, -2, \dots\}. Another basic case occurs when a = b, yielding M(a, a, z) = e^{z} for all finite z \in \mathbb{C} and a \notin \{0, -1, -2, \dots\}. These identities follow directly from the defining power series expansion. The following table summarizes these and related common evaluations:| Parameters | Expression |
|---|---|
| a = 0, b \notin \{0, -1, -2, \dots\}, any z | $1 |
| any a, b \notin \{0, -1, -2, \dots\}, z = 0 | $1 |
| a = b \notin \{0, -1, -2, \dots\}, any z | e^{z} |