Hermite polynomials
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: Hermite polynomials were defined by Pierre-Simon Laplace in 1810, though in scarcely recognizable form, and studied in detail by Pafnuty Chebyshev in 1859. Chebyshev's work was overlooked, and they were named later after Charles Hermite, who wrote on the polynomials in 1864, describing them as new. They were consequently not new, although Hermite was the first to define the multidimensional polynomials in his later 1865 publications. Definition. Like the other classical orthogonal polynomials, the Hermite polynomials can be defined from several  different starting points. Noting from the outset that there are two different standardizations in common use, one convenient method is as follows: These equations have the form of a Rodrigues' formula and can also be written as, formula_4 The two definitions are not exactly identical; each is a rescaling of the other: formula_5 These are Hermite polynomial sequences of different variances; see the material on variances below. The notation and is that used in the standard references. The polynomials are sometimes denoted by , especially in probability theory, because formula_6 is the probability density function for the normal distribution with expected  value 0 and standard deviation 1. Properties. The th-order Hermite polynomial is a polynomial of degree . The probabilist's version has leading coefficient 1, while the physicist's version has leading coefficient . Symmetry. From the Rodrigues formulae given above, we can see that and are even or odd functions depending on : formula_9 Orthogonality. and are th-degree polynomials for . These polynomials are orthogonal with respect to the "weight function" (measure) formula_10 or formula_11 i.e., we have formula_12 Furthermore, formula_13 or formula_14 where formula_15 is the Kronecker delta. The probabilist polynomials are thus orthogonal with respect to the standard normal  probability density function. Completeness. The Hermite polynomials (probabilist's or physicist's) form an orthogonal basis of the Hilbert space of functions satisfying formula_16 in which the inner product is given by the integral formula_17 including the Gaussian weight function defined in the preceding section An orthogonal basis for is a "complete" orthogonal system. For an orthogonal system, "completeness" is equivalent to the fact that the 0 function is the only function orthogonal to "all" functions in the system. Since the linear span of Hermite polynomials is the space of all polynomials, one has to show (in physicist case) that  if satisfies formula_18 for every , then . One possible way to do this is to appreciate that the entire function formula_19 vanishes identically. The fact then that for every real means that the Fourier transform of is 0, hence is 0 almost everywhere. Variants of the above completeness proof apply to other weights with exponential decay. In the Hermite case, it is also possible to prove an explicit identity that implies completeness (see section on the Completeness relation below). An equivalent formulation of the fact that Hermite polynomials are an orthogonal basis for consists in introducing Hermite "functions" (see below), and  in saying that the Hermite functions are an orthonormal basis for . Hermite's differential equation. The probabilist's Hermite polynomials are solutions of the differential equation formula_20 where is a constant. Imposing the boundary condition that should be polynomially bounded at infinity, the equation has solutions only if is a non-negative integer, and the solution is uniquely given by formula_21, where formula_22 denotes a constant. Rewriting the differential equation as an eigenvalue problem formula_23 the Hermite polynomials formula_24 may be understood as eigenfunctions of the differential operator formula_25 . This eigenvalue problem is called the Hermite equation, although the term  is also used for the closely related equation formula_26 whose solution is uniquely given in terms of physicist's Hermite polynomials in the form formula_27, where formula_22 denotes a constant, after imposing the boundary condition that should be polynomially bounded at infinity. The general solutions to the above second-order differential equations are in fact linear combinations of both Hermite polynomials and confluent hypergeometric functions of the first kind. For example, for the physicist's Hermite equation formula_29 the general solution takes the form formula_30 where formula_22 and formula_32 are constants, formula_33 are physicist's Hermite polynomials (of the first  kind), and formula_34 are physicist's Hermite functions (of the second kind). The latter functions are compactly represented as formula_35 where formula_36 are Confluent hypergeometric functions of the first kind. The conventional Hermite polynomials may also be expressed in terms of confluent hypergeometric functions, see below. With more general boundary conditions, the Hermite polynomials can be generalized to obtain more general analytic functions for complex-valued . An explicit formula of Hermite polynomials in terms of contour integrals is also possible. Recurrence relation. The sequence of probabilist's Hermite polynomials also satisfies the recurrence relation formula_37 Individual coefficients are related by  the following recursion formula: formula_38 and , , . For the physicist's polynomials, assuming formula_39 we have formula_40 Individual coefficients are related by the following recursion formula: formula_41 and , , . The Hermite polynomials constitute an Appell sequence, i.e., they are a polynomial sequence satisfying the identity formula_42 Equivalently, by Taylor-expanding, formula_43 These umbral identities are self-evident and included in the differential operator representation detailed below, formula_44 In consequence, for the th derivatives the following relations hold: formula_45 It follows that the Hermite polynomials also satisfy the recurrence relation formula_46 These last relations, together with the initial polynomials and , can be used  in practice to compute the polynomials quickly. Turán's inequalities are formula_47 Moreover, the following multiplication theorem holds: formula_48 Explicit expression. The physicist's Hermite polynomials can be written explicitly as formula_49 These two equations may be combined into one using the floor function: formula_50 The probabilist's Hermite polynomials have similar formulas, which may be obtained from these by replacing the power of with the corresponding power of and multiplying the entire sum by : formula_51 Inverse explicit expression. The inverse of the above explicit expressions, that is, those for monomials in terms of probabilist's Hermite polynomials are formula_52  The corresponding expressions for the physicist's Hermite polynomials follow directly by properly scaling this: formula_53 Generating function. The Hermite polynomials are given by the exponential generating function formula_54 This equality is valid for all complex values of and , and can be obtained by writing the Taylor expansion at of the entire function (in the physicist's case). One can also derive the (physicist's) generating function by using Cauchy's integral formula to write the Hermite polynomials as formula_55 Using this in the sum formula_56 one can evaluate the remaining integral using the calculus of residues and arrive at  the desired generating function. Expected values. If is a random variable with a normal distribution with standard deviation 1 and expected value , then formula_57 The moments of the standard normal (with expected value zero) may be read off directly from the relation for even indices: formula_58 where is the double factorial. Note that the above expression is a special case of the representation of the probabilist's Hermite polynomials as moments: formula_59 Asymptotic expansion. Asymptotically, as , the expansion formula_60 holds true. For certain cases concerning a wider range of evaluation, it is necessary to include a factor for changing amplitude : formula_61 which, using Stirling's approximation, can be further simplified, in the limit, to formula_62 This expansion is needed to resolve the wavefunction of a quantum harmonic oscillator such that it agrees with the classical approximation in the limit of the correspondence principle. A better approximation, which accounts for the variation in frequency, is given by formula_63 A finer approximation, which takes into account the uneven spacing of the zeros near the edges, makes use of the substitution formula_64 with which one has the uniform approximation formula_65 Similar approximations hold for the monotonic and transition regions. Specifically, if formula_66 then  formula_67 while for formula_68 with complex and bounded, the approximation is formula_69 where is the Airy function of the first kind. Special values. The physicist's Hermite polynomials evaluated at zero argument are called Hermite numbers. formula_70 which satisfy the recursion relation . In terms of the probabilist's polynomials this translates to formula_71 Relations to other functions. Laguerre polynomials. The Hermite polynomials can be expressed as a special case of the Laguerre polynomials: formula_72 Relation to confluent hypergeometric functions. The physicist's Hermite polynomials can be expressed as a special case of the parabolic cylinder functions: formula_73 in the right  half-plane, where is Tricomi's confluent hypergeometric function. Similarly, formula_74 where is Kummer's confluent hypergeometric function. Differential-operator representation. The probabilist's Hermite polynomials satisfy the identity formula_75 where represents differentiation with respect to , and the exponential is interpreted by expanding it as a power series. There are no delicate questions of convergence of this series when it operates on polynomials, since all but finitely many terms vanish. Since the power-series coefficients of the exponential are well known, and higher-order derivatives of the monomial can be written down explicitly, this differential-operator representation gives rise to  a concrete formula for the coefficients of that can be used to quickly compute these polynomials. Since the formal expression for the Weierstrass transform is , we see that the Weierstrass transform of is . Essentially the Weierstrass transform thus turns a series of Hermite polynomials into a corresponding Maclaurin series. The existence of some formal power series with nonzero constant coefficient, such that , is another equivalent to the statement that these polynomials form an Appell sequence. Since they are an Appell sequence, they are "a fortiori" a Sheffer sequence. Contour-integral representation. From the generating-function representation above, we see that  the Hermite polynomials have a representation in terms of a contour integral, as formula_76 with the contour encircling the origin. Generalizations. The probabilist's Hermite polynomials defined above are orthogonal with respect to the standard normal probability distribution, whose density function is formula_77 which has expected value 0 and variance 1. Scaling, one may analogously speak of generalized Hermite polynomials formula_78 of variance , where is any positive number. These are then orthogonal with respect to the normal probability distribution whose density function is formula_79 They are given by formula_80 Now, if formula_81 then the polynomial sequence whose th term is  formula_82 is called the umbral composition of the two polynomial sequences. It can be shown to satisfy the identities formula_83 and formula_84 The last identity is expressed by saying that this parameterized family of polynomial sequences is known as a cross-sequence. (See the above section on Appell sequences and on the differential-operator representation, which leads to a ready derivation of it. This binomial type identity, for , has already been encountered in the above section on #Recursion relations.) Section::::"Negative variance". Since polynomial sequences form a group under the operation of umbral composition, one may denote by formula_85 the sequence  that is inverse to the one similarly denoted, but without the minus sign, and thus speak of Hermite polynomials of negative variance. For , the coefficients of formula_86 are just the absolute values of the corresponding coefficients of formula_87. These arise as moments of normal probability distributions: The th moment of the normal distribution with expected value and variance is formula_88 where is a random variable with the specified normal distribution. A special case of the cross-sequence identity then says that formula_89 Applications. Hermite functions. One can define the Hermite functions (often called Hermite-Gaussian functions) from the physicist's  polynomials: formula_90 Thus, formula_91 Since these functions contain the square root of the weight function and have been scaled appropriately, they are orthonormal: formula_92 and they form an orthonormal basis of . This fact is equivalent to the corresponding statement for Hermite polynomials (see above). The Hermite functions are closely related to the Whittaker function : formula_93 and thereby to other parabolic cylinder functions. The Hermite functions satisfy the differential equation formula_94 This equation is equivalent to the Schrödinger equation for a harmonic oscillator in quantum mechanics, so these functions are the eigenfunctions. formula_95 Recursion relation. Following recursion relations of Hermite polynomials , the Hermite functions obey formula_96 and formula_97 Extending the first relation to the arbitrary th derivatives for any positive integer leads to formula_98 This formula can be used in connection with the recurrence relations for and to calculate any derivative of the Hermite functions efficiently. Cramér's inequality. For real , the Hermite functions satisfy the following bound due to Harald Cramér and Jack Indritz: formula_99 Hermite functions as eigenfunctions of the Fourier transform. The Hermite functions are a set of eigenfunctions of the continuous Fourier transform . To see this, take the physicist's version of the generating function and multiply  by . This gives formula_100 The Fourier transform of the left side is given by formula_101 The Fourier transform of the right side is given by formula_102 Equating like powers of in the transformed versions of the left and right sides finally yields formula_103 The Hermite functions are thus an orthonormal basis of , which "diagonalizes the Fourier transform operator". Wigner distributions of Hermite functions. The Wigner distribution function of the th-order Hermite function is related to the th-order Laguerre polynomial. The Laguerre polynomials are formula_104 leading to the oscillator Laguerre functions formula_105 For all natural integers , it is straightforward  to see that formula_106 where the Wigner distribution of a function is defined as formula_107 This is a fundamental result for the quantum harmonic oscillator discovered by Hip Groenewold in 1946 in his PhD thesis. It is the standard paradigm of quantum mechanics in phase space. There are further relations between the two families of polynomials. Combinatorial interpretation of coefficients. In the Hermite polynomial of variance 1, the absolute value of the coefficient of is the number of (unordered) partitions of an -element set into singletons and (unordered) pairs. Equivalently, it is the number of involutions of an -element set  with precisely fixed points, or in other words, the number of matchings in the complete graph on vertices that leave vertices uncovered (indeed, the Hermite polynomials are the matching polynomials of these graphs). The sum of the absolute values of the coefficients gives the total number of partitions into singletons and pairs, the so-called telephone numbers This combinatorial interpretation can be related to complete exponential Bell polynomials as formula_108 where for all . These numbers may also be expressed as a special value of the Hermite polynomials: formula_109 Completeness relation. The Christoffel–Darboux formula for Hermite polynomials reads formula_110 Moreover , the following completeness identity for the above Hermite functions holds in the sense of distributions: formula_111 where is the Dirac delta function, the Hermite functions, and represents the Lebesgue measure on the line in , normalized so that its projection on the horizontal axis is the usual Lebesgue measure. This distributional identity follows by taking in Mehler's formula, valid when : formula_112 which is often stated equivalently as a separable kernel, formula_113 The function is the bivariate Gaussian probability density on , which is, when is close to 1, very concentrated around the line , and very spread out on that line. It  follows that formula_114 when and are continuous and compactly supported. This yields that can be expressed in Hermite functions as the sum of a series of vectors in , namely, formula_115 In order to prove the above equality for , the Fourier transform of Gaussian functions is used repeatedly: formula_116 The Hermite polynomial is then represented as formula_117 With this representation for and , it is evident that formula_118 and this yields the desired resolution of the identity result, using again the Fourier transform of Gaussian kernels under the substitution formula_119