17.2. LINEAR DIFFERENCE EQUATIONS WITH A SINGLE CONTINUOUS VARIABLE 899 17.2.3-2. Linear nonhomogeneous difference equations with constant coefficients. An mth-order linear nonhomogeneous integer-difference equation with constant coeffi- cients has the form a m y(x + m)+a m–1 y(x + m – 1)+···+ a 1 y(x + 1)+a 0 y(x)=f (x), (17.2.3.5) where a 0 a m 0. The general solution of the nonhomogeneous equation (17.2.3.5) is given by the sum y(x)=u(x)+y(x), where u(x) is the general solution of the corresponding homogeneous equation (with f ≡ 0), and y(x) is a particular solution of the nonhomogeneous equation (17.2.3.5). Regarding the solution of the homogeneous equation see Paragraph 17.2.3-1. Below, we consider some methods for the construction of a particular solution of the nonhomogeneous equation (17.2.3.5). 1 ◦ . Table 17.1 lists the forms of particular solutions corresponding to some special cases of the function on the right-hand side of the linear nonhomogeneous difference equation (17.2.3.5). 2 ◦ .Let P (λ)=a m λ m + a m–1 λ m–1 + ···+ a 1 λ + a 0 be the characteristic polynomial of equation (17.2.3.5). Consider the function g(λ)= 1 P (λ) = ∞ k=0 g k λ k , |λ| < |λ 1 |,(17.2.3.6) where λ 1 is the root of the equation P (λ)=0 with the smallest absolute value. Then lim k→∞ |g k | 1/k = |λ 1 | –1 . From (17.2.3.6), it follows that a 0 g 0 = 1, a 1 g 0 + a 0 g 1 = 0, , s–1 k=0 a s–k–1 g k = 0, s ≥ 2, a ν = 0 if ν > m. If lim k→∞ |f(x + k)| 1/k = σ f < |λ 1 |, then the series y(x)=g 0 f(x)+g 1 f(x + 1)+···+ g k f(x + k)+··· is convergent and its sum gives a solution of equation (17.2.3.5). 3 ◦ . Let the right-hand side of equation (17.2.3.5) can be represented by the integral f(x)= L F (λ)λ x–1 dλ,(17.2.3.7) where the line of integration L does not cross the roots of the characteristic polynomial P (λ). Direct verification shows that the integral y(x)= L F (λ) P (λ) λ x–1 dλ (17.2.3.8) represents a particular solution of equation (17.2.3.5). 900 DIFFERENCE EQUATIONS AND OTHER FUNCTIONAL EQUATIONS TABLE 17.1 Forms of particular solutions of the linear nonhomogeneous difference equation with constant coefficients a m y(x + m)+a m–1 y(x + m – 1)+···+ a 1 y(x + 1)+a 0 y(x)=f(x) in some special cases of the function f (x); a 0 a m 0 Form of the function f (x) Roots of the characteristic equation a m λ m + a m–1 λ m–1 + ···+ a 1 λ + a 0 = 0 Form of a particular solution y = y(x) λ = 1 is not a root of the characteristic equation (i.e., a m + a m–1 + ···+ a 1 + a 0 ≠ 0) n k=0 b k x k x n λ = 1 is a root of the characteristic equation (multiplicity r) n+r k=0 b k x k λ = e β is not a root of the characteristic equation be βx e βx (β is a real constant) λ = e β is a root of the characteristic equation (multiplicity r) e βx r k=0 b k x k β is not a root of the characteristic equation e βx n k=0 b k x k x n e βx (β is a real constant) β is a root of the characteristic equation (multiplicity r) e βx n+r k=0 b k x k iβ is not a root of the characteristic equation P ν (x)cosβx + Q ν (x)sinβx P m (x)cosβx + Q n (x)sinβx iβ is a root of the characteristic equation (multiplicity r) P ν+r (x)cosβx + Q ν+r (x)sinβx α + iβ is not a root of the characteristic equation [ P ν (x)cosβx + Q ν (x)sinβx]e αx [P m (x)cosβx + Q n (x)sinβx]e αx α + iβ is a root of the characteristic equation (multiplicity r) [ P ν+r (x)cosβx + Q ν+r (x)sinβx]e αx Notation: P m (x)andQ n (x) are polynomials of degrees m and n with given coefficients; P m (x), P ν (x), and Q ν (x) are polynomials of degrees m and ν whose coefficients are determined by substituting the particular solution into the basic equation; ν =max(m, n); and α and β are real numbers, i 2 =–1. Example 1. Taking F (λ)=aλ b , b ≥ 0,andL = {0 ≤ x ≤ 1} in (17.2.3.7), we have f(x)= 1 0 aλ b λ x–1 dλ = a x + b . Therefore, if the characteristic polynomial P (λ) has no roots on the segment [0, 1], then a particular solution of equation (17.2.3.5) with the right-hand side f(x)= a x + b (17.2.3.9) can be obtained in the form y(x)=a 1 0 λ b P (λ) λ x–1 dλ. (17.2.3.10) Remark. For L = {0 ≤ x < ∞}, the representation (17.2.3.7) may be regarded as the Mellin transformation that maps F (λ)intof(x). 4 ◦ .Lety(x) be a solution of equation (17.2.3.5). Then u(x)=βy x (x) is a solution of the nonhomogeneous equation a m u(x + m)+a m–1 u(x + m – 1)+···+ a 1 u(x + 1)+a 0 u(x)=βf x (x). 17.2. LINEAR DIFFERENCE EQUATIONS WITH A SINGLE CONTINUOUS VARIABLE 901 Example 2. In order to find a solution of equation (17.2.3.5) with the right-hand side f(x)= a (x + b) 2 =– d dx a x + b , (17.2.3.11) let us multiply (17.2.3.9)–(17.2.3.10) by β =–1 and then differentiate the resulting expressions. Thus, we obtain the following particular solution of the nonhomogeneous equation (17.2.3.5) with the right-hand side (17.2.3.11): y(x)=–a 1 0 ln λ P (λ) λ x+b–1 dλ. (17.2.3.12) Consecutively differentiating expressions (17.2.3.11) and (17.2.3.12), we obtain a solution of equation (17.2.3.5) with the right-hand side f(x)= a (x + b) n = (–1) n–1 (n – 1)! d n–1 dx n–1 a x + b . This solution has the form y(x)=a (–1) n–1 (n – 1)! 1 0 (ln λ) n–1 P (λ) λ x+b–1 dλ. 5 ◦ . In Paragraph 17.2.3-4, Item 3 ◦ , there is a formula that allows us to obtain a particular solution of the nonhomogeneous equation (17.2.3.5) with an arbitrary right-hand side. 17.2.3-3. Linear homogeneous difference equations with variable coefficients. 1 ◦ .Anmth-order linear homogeneous integer-difference equation with variable coeffi- cients has the form a m (x)y(x + m)+a m–1 (x)y(x + m – 1)+···+ a 1 (x)y(x + 1)+a 0 (x)y(x)=0,(17.2.3.13) where a 0 (x)a m (x) 0. This equation admits the trivial solution y(x) ≡ 0. The set E of all singular points of equation (17.2.3.13) consists of points of three classes: 1) zeroes of the function a 0 (x), denoted by μ 1 , μ 2 , ; 2) zeroes of the function a m (x – m), denoted by ν 1 , ν 2 , ; 3) singular points of the coefficients of the equation, denoted by η 1 , η 2 , The points of the set S(E)={μ s – n, ν s + n, η s – n, η s + m + n; n = 0, 1, 2, , s = 1, 2, 3, } are called comparable with singular points of equation (17.2.3.13). Let y 1 = y 1 (x), y 2 = y 2 (x), , y m = y m (x)(17.2.3.14) be particular solutions of equation (17.2.3.13). Then the function y = Θ 1 (x)y 1 (x)+Θ 2 (x)y 2 (x)+···+ Θ m (x)y m (x)(17.2.3.15) with arbitrary 1-periodic functions Θ 1 (x), Θ 2 (x), , Θ m (x) is also a solution of equation (17.2.3.13). The Casoratti determinant is the function defi ned as D(x)= y 1 (x) y 2 (x) ··· y m (x) y 1 (x + 1) y 2 (x + 1) ··· y m (x + 1) ··· ··· ··· ··· y 1 (x + m – 1) y 2 (x + m – 1) ··· y m (x + m – 1) .(17.2.3.16) 902 DIFFERENCE EQUATIONS AND OTHER FUNCTIONAL EQUATIONS THEOREM (CASORATTI). Formula (17.2.3.15) gives the general solution of the linear homogeneous difference equation (17.2.3.13) if and only if for any point x S(E) such that x + k S(E) for k = 0, 1, , m , the condition D(x) ≠ 0 holds. The Casoratti determinant (17.2.3.16) satisfies the first-order difference equation D(x + 1)=(–1) m a 0 (x) a m (x) D(x). (17.2.3.17) 2 ◦ .Lety 0 = y 0 (x) be a nontrivial particular solution of equation (17.2.3.13). Then the order of equation (17.2.3.13) can be reduced by unity. Indeed, making the replacement y(x)=y 0 (x)u(x)(17.2.3.18) in equation (17.2.3.13), we get m k=0 a k (x)y 0 (x + k)u(x + k)=0.(17.2.3.19) Let us transform this relation with the help of the Abel identity m k=0 F k G k =– m–1 k=0 (G k+1 – G k ) k s=0 F s + G m m k=0 F k , in which we take F k = a k (x)y 0 (x + k)andG k = u(x + k). As a result, we get – m–1 k=0 r k (x)[u(x + k + 1)–u(x + k)] + u(x + m) m k=0 a k (x)y 0 (x + k)=0,(17.2.3.20) where r k (x)= k s=0 a s (x)y 0 (x + s), k = 0, 1, , m – 1. Taking into account that the second sum in (17.2.3.20) is equal to zero (since y 0 is a particular solution of the equation under consideration) and setting w(x)=u(x + 1)–u(x)(17.2.3.21) in (17.2.3.20), we come to an (m – 1)th-order difference equation m–1 k=0 r k (x)w(x + k)=0. 17.2.3-4. Linear nonhomogeneous difference equations with variable coefficients. 1 ◦ .Anmth-order linear nonhomogeneous integer-difference equation with variable coef- ficients has the form a m (x)y(x +m)+a m–1 (x)y(x +m –1)+···+a 1 (x)y(x +1)+a 0 (x)y(x)=f(x), (17.2.3.22) where a 0 (x)a m (x) 0. The general solution of the nonhomogeneous equation (17.2.3.22) is given by the sum y(x)=u(x)+y(x), (17.2.3.23) where u(x) is the general solution of the corresponding homogeneous equation (with f ≡ 0) and y(x) is a particular solution of the nonhomogeneous equation (17.2.3.22). The general solution of the homogeneous equation is defined by the right-hand side of (17.2.3.15). Every solution of equation (17.2.3.22) is uniquely determined by prescribing the values of the sought function on the interval [0, m). 17.2. LINEAR DIFFERENCE EQUATIONS WITH A SINGLE CONTINUOUS VARIABLE 903 2 ◦ . A particular solution y(x) of the linear nonhomogeneous difference equation a m (x)y(x + m)+a m–1 (x)y(x + m – 1)+···+ a 1 (x)y(x + 1)+a 0 (x)y(x)= n k=1 f k (x) can be represented by the sum y(x)= n k=1 y k (x), where y k (x) are particular solutions of the linear nonhomogeneous difference equations a m (x)y(x + m)+a m–1 (x)y(x + m – 1)+···+ a 1 (x)y(x + 1)+a 0 (x)y(x)=f k (x). 3 ◦ . The solution of the Cauchy problem for the nonhomogeneous equation (17.2.3.22) with arbitrary initial conditions y(x + j)=ϕ j (x)for0 ≤ x < 1, j = 0, 1, , m – 1,(17.2.3.24) is given by the sum (17.2.3.23), where u(x)=– 1 D({x}) 0 y 1 (x) ··· y m (x) ϕ 0 ({x}) y 1 ({x}) ··· y m ({x}) ··· ··· ··· ··· ϕ m–1 ({x}) y 1 ({x} + m – 1) ··· y m ({x} + m – 1) is a solution of the homogeneous equation (17.2.3.13) with the boundary conditions (17.2.3.24), and y(x)= [x] j=m D ∗ (x – j + 1) D(x – j + 1) f(x – j) a m (x – j) (17.2.3.25) is a particular evolution of the nonhomogeneous equation (17.2.3.22) with zero initial conditions y(x)=0 for 0 ≤ x < m. Formula (17.2.3.25) contains the determinant D ∗ (t + 1)= y 1 (t + 1) ··· y m (t + 1) ··· ··· ··· y 1 (t + m – 1) ··· y m (t + m – 1) y 1 (x) ··· y m (x) , which is obtained from the determinant D(t + 1) by replacing the last row [y 1 (t + m), , y m (t+m)] with the row [y 1 (x), , y m (x)]. Note that D ∗ (t+1)=0 for x–m+1 ≤ t≤x–1. 17.2.3-5. Equations reducible to equations with constant coefficients. 1 ◦ . The difference equation with variable coefficients a m f(x + m)y(x + m)+a m–1 f(x + m – 1)y(x + m – 1)+··· + a 1 f(x + 1)y(x + 1)+a 0 f(x)y(x)=g(x) can be reduced, with the help of the replacement y(x)=f (x)u(x), 904 DIFFERENCE EQUATIONS AND OTHER FUNCTIONAL EQUATIONS to the equation with constant coefficients a m u(x + m)+a m–1 u(x + m – 1)+···+ a 1 u(x + 1)+a 0 u(x)=g(x). 2 ◦ . The difference equation with variable coefficients a m y(x + m)+a m–1 f(x)y(x + m – 1)+a m–2 f(x)f (x – 1)y(x + m – 2)+··· + a 0 f(x)f (x – 1) f(x – m + 1)y(x)=g(x) can be reduced to a nonhomogeneous equation with constant coefficients with the help of the replacement y(x)=u(x)exp ϕ(x – m) , with ϕ(x) satisfying the auxiliary first-order difference equation ϕ(x + 1)–ϕ(x)=lnf(x). The resulting equation with constant coefficients has the form a m u(x + m)+a m–1 u(x – m – 1)+···+ a 1 u(x + 1)+a 0 u(x)=g(x)exp[–ϕ(x)]. 3 ◦ . The difference equation with variable coefficients a m f(x)f (x+1) f(x+m–1)y(x+m)+a m–1 f(x)f (x+1) f(x+m–2)y(x+m–1)+··· + a 1 f(x)y(x + 1)+a 0 y(x)=g(x) can be reduced to a nonhomogeneous equation with constant coefficients with the help of the replacement y(x)=u(x)exp –ψ(x) , where ϕ(x) is a function satisfying the auxiliary first-order difference equation ψ(x + 1)–ψ(x)=lnf(x). The resulting equation with constant coefficients has the form a m u(x + m)+a m–1 u(x – m – 1)+···+ a 1 u(x + 1)+a 0 u(x)=g(x)exp[ψ(x)]. 17.2.4. Linear mth-Order Difference Equations with Arbitrary Differences 17.2.4-1. Linear homogeneous difference equations. 1 ◦ .Alinear homogeneous difference equation with constant coefficients, in the case of arbitrary differences, has the form a m y(x + h m )+a m–1 y(x + h m–1 )+···+ a 1 y(x + h 1 )+a 0 y(x + h 0 )=0,(17.2.4.1) where a 0 a m ≠ 0, m ≥ 1, h i ≠ h j for i ≠ j; the coefficients a k and the differences h k are complex numbers and x is a complex variable. Equation (17.2.4.1) can be reduced to an equation with integer differences if the quan- tities h k – h 0 are commensurable in the sense that there is a common constant q such that h k – h 0 = qN k with integer N k . Indeed, we have y(x + h k )=y(x + h 0 + qN k )=y q x + h 0 q + N k = w(z + N k ), where the new variables have the form z =(x + h 0 )/q and w(z)=y(qz). As a result, we obtain an equation with integer differences N k for the function w(z). In particular, this situation takes place if h k – h 0 are rational numbers: h k – h 0 = p k /r k ,wherep k and r k are positive integers (k = 1, 2, , m). In this case, one can take q = 1/r,wherer is the common denominator of the fractions p k /r k . 17.2. LINEAR DIFFERENCE EQUATIONS WITH A SINGLE CONTINUOUS VARIABLE 905 2 ◦ . In what follows, we assume that the numbers h k – h 0 are not commensurable (k = 1, , m). We seek particular solutions of equation (17.2.4.1) in the form y = e tx . Substituting this expression into (17.2.4.1) and dividing the result by e tx , we obtain the transcendental equation A(t) ≡ a m e h m t + a m–1 e h m–1 t + ···+ a 1 e h 1 t + a 0 e h 0 t = 0,(17.2.4.2) where A(t) is the characteristic function. It is known that equation (17.2.4.2) has infinitely many roots. 3 ◦ .Leth k be real ordered numbers, h 0 < h 1 < ··· < h m ,andt = t 1 + it 2 ,wheret 1 =Ret and t 2 =Imt. Then the following statements hold: (a) There exist constants γ 1 and γ 2 such that the following estimates are valid: |A(t)| > 1 2 |a 0 |e h 0 t 1 if t 1 ≤ γ 1 , |A(t)| > 1 2 |a m |e h m t 1 if t 1 ≥ γ 2 . (b) All roots of equation (17.2.4.2) belong to the vertical strip γ 1 < t 1 < γ 2 . (c) Let β 1 , β 2 , , β n , be roots of equation (17.2.4.2), and |β 1 | ≤ |β 2 | ≤ ···≤ |β n | ≤ ··· Then the following limit relation holds: lim n→∞ β n n = 2π h m – h 0 . 4 ◦ . A root β k of multiplicity n k of the characteristic equation (17.2.4.2) corresponds to exactly n k linearly independent solutions of equation (17.2.4.1): e β k x , xe β k x , , x n k –1 e β k x (k = 1, 2, ). (17.2.4.3) Equation (17.2.4.1) admits infinitely many solutions of the form (17.2.4.3), since the characteristic function (17.2.4.2) has infinitely many roots. In order to single out different classes of solutions, it is convenient to use a condition that characterizes the order of their growth for large values of the argument. The class of functions of exponential growth of finite degree σ is denoted by [1, σ]and is defined as the set of all entire functions* f(x) satisfying the condition e (σ–ε)|x| < |f(x)| < e (σ+ε)|x| (17.2.4.4) for any ε > 0, where the right inequality in (17.2.4.4) should hold for all sufficiently large x: |x| > R(ε), while the left inequality in (17.2.4.4) should hold for some sequence x = x n = x n (ε) →∞. The parameter σ can be found from the relations σ = lim r→∞ M(r) r = lim k→∞ k! |b k | 1/k , M(r)=max |x|=r |f(x)| (0 ≤ σ < ∞), where b k are the coefficients in the power series expansion of the function f (x)(seethe footnote). * An entire function is a function that is analytic on the entire complex plane (except, possibly, the infinite point). Any such function can be expanded in power series, f(x)= ∞ k=0 b k x k , convergent on the entire complex plane, i.e., lim k→∞ |b k | 1/k = 0. . methods for the construction of a particular solution of the nonhomogeneous equation (17.2.3.5). 1 ◦ . Table 17.1 lists the forms of particular solutions corresponding to some special cases of the. characterizes the order of their growth for large values of the argument. The class of functions of exponential growth of finite degree σ is denoted by [1, σ ]and is defined as the set of all entire functions*. some special cases of the function f (x); a 0 a m 0 Form of the function f (x) Roots of the characteristic equation a m λ m + a m–1 λ m–1 + ···+ a 1 λ + a 0 = 0 Form of a particular solution