934 DIFFERENCE EQUATIONS AND OTHER FUNCTIONAL EQUATIONS where B 1 , , B 4 are arbitrary constants and k 1 and k 2 are the roots of the quadratic equation (k – A 1 )(k – A 4 )–A 2 A 3 = 0.(17.5.5.23) In the degenerate case k 1 = k 2 ,thetermse k 2 ϕ and e –k 2 z in (17.5.5.22) should be replaced with ϕe k 1 ϕ and ze –k 1 z , respectively. In the case of purely imaginary or complex roots, one should separate the real (or imaginary) part of the roots in solution (17.5.5.22). On substituting (17.5.5.22) into the original functional equation, one obtains conditions for the free coefficients and identifies the function f(t), namely, B 2 = B 4 = 0 =⇒ f(t)=[A 2 A 3 +(k 1 –A 1 ) 2 ]B 1 B 3 e –k 1 ψ , B 1 = B 3 = 0 =⇒ f(t)=[A 2 A 3 +(k 2 –A 1 ) 2 ]B 2 B 4 e –k 2 ψ , A 1 = 0 =⇒ f(t)=(A 2 A 3 +k 2 1 )B 1 B 3 e –k 1 ψ +(A 2 A 3 +k 2 2 )B 2 B 4 e –k 2 ψ . (17.5.5.24) Solution (17.5.5.22), (17.5.5.24) involves arbitrary functions ϕ = ϕ(x)andψ = ψ(t). Degenerate case. In addition, the functional equation has two degenerate solutions [formulas (17.5.5.7) are used], f = B 1 B 2 e A 1 ψ , g = A 2 B 1 e –A 1 ϕ , h = B 1 e –A 1 ϕ , R =–B 2 e A 1 z – A 2 Q, where ϕ = ϕ(x), ψ = ψ(t), and Q = Q(z) are arbitrary functions; A 1 , A 2 , B 1 ,andB 2 are arbitrary constants; and f = B 1 B 2 e A 1 ψ , h =–B 1 e –A 1 ϕ – A 2 g, Q = A 2 B 2 e A 1 z , R = B 2 e A 1 z , where ϕ = ϕ(x), ψ = ψ(t), and g = g(x) are arbitrary functions; and A 1 , A 2 , B 1 ,andB 2 are arbitrary constants. 3 ◦ . Consider a more general functional equation of the form f(t)+g 1 (x)Q 1 (z)+···+ g n (x)Q n (z)=0,wherez = ϕ(x)+ψ(t). (17.5.5.25) By differentiation in x, this equation can be reduced to a functional differential equation, which may be regarded as a bilinear functional equation of the form (17.5.5.8). Using formulas (17.5.5.9) for the construction of its solution, one can first obtain a system of ODEs and then find solutions of the original equation (17.5.5.25). 4 ◦ . Consider a functional equation of the form f 1 (t)g 1 (x)+···+ f m (t)g m (x)+h 1 (x)Q 1 (z)+···+ h n (x)Q n (z)=0, z = ϕ(x)+ψ(t). (17.5.5.26) Assume that g m (x) 0. Dividing equation (17.5.5.26) by g m (x) and differentiating the result in x, we come to an equation of the form f 1 (t)¯g 1 (x)+···+ f m–1 (t)¯g m–1 (x)+ 2n  i=1 s i (x)R i (z)=0 with a smaller number of functions f i (t). Proceeding in this way, we can eliminate all functions f i (t) and obtain a functional-differential equation with two variables of the form (17.5.5.10)–(17.5.5.11), which can be reduced to the standard bilinear functional equation by the method of splitting. REFERENCES FOR CHAPTER 17 935 17.5.5-4. Nonlinear equations containing the complex argument z = ϕ(t)θ(x)+ψ(t). Consider a functional equation of the form [α 1 (t)θ(x)+β 1 (t)]R 1 (z)+···+[α n (t)θ(x)+β n (t)]R n (z)=0, z = ϕ(t)θ(x)+ψ(t). (17.5.5.27) Passing in (17.5.5.27) from the variables x and t to new variables z and t [the function θ is replaced by (z – ψ)/ϕ], we come to the bilinear equation of the form (17.5.5.8): n  i=1 α i (t)zR i (z)+ n  i=1 [ϕ(t)β i (t)–ψ(t)α i (t)]R i (z)=0. Remark. Instead of the expressions α i (t)θ(x)+β i (t) in (17.5.5.27) linearly depending on the function θ(x), one can consider polynomials of θ(x) with coefficients depending on t.  See also Section T12.3 for exact solutions of some linear and nonlinear difference and functional equations with several independent variables. References for Chapter 17 Acz ´ el, J., Functional Equations: History, Applications and Theory, Kluwer Academic, Dordrecht, 2002. Acz ´ el, J., Lectures on Functional Equations and Their Applications, Dover Publications, New York, 2006. Acz ´ el, J., Some general methods in the theory of functional equations with a single variable. New applications of functional equations [in Russian], Uspekhi Mat. Nauk, Vol. 11, No. 3 (69), pp. 3–68, 1956. Acz ´ el, J. and Dhombres, J., Functional Equations in Several Variables, Cambridge University Press, Cam- bridge, 1989. Agarwal, R. P., Difference Equations and Inequalities, 2nd Edition, Marcel Dekker, New York, 2000. Balasubrahmanyan, R. and Lau, K., Functional Equations in Probability Theory, Academic Press, San Diego, 1991. Belitskii, G. R. and Tkachenko,V.,One-Dimensional Functional Equations,Birkh ¨ auser Verlag, Boston, 2003. Castillo, E. and Ruiz-Cobo, M. R., Functional Equations and Modelling in Science and Engineering, Marcel Dekker, New York, 1992. Castillo, E. and Ruiz-Cobo, R., Functional Equations in Applied Sciences, Elsevier, New York, 2005. Czerwik, S., Functional Equations and Inequalities in Several Variables, World Scientific Publishing Co., Singapore, 2002. Daroczy, Z. and Pales, Z. (Editors), Functional Equations — Results and Advances, Kluwer Academic, Dordrecht, 2002. Eichhorn, W., Functional Equations in Economics, Addison Wesley, Reading, Massachusetts, 1978. Elaydi, S., An Introduction to Difference Equations, 3rd Edition, Springer-Verlag, New York, 2005. Goldberg, S., Introduction to Difference Equations, Dover Publications, New York, 1986. Jarai, A., Regularity Properties of Functional Equations in Several Variables, Springer-Verlag, New York, 2005. Kelley, W. and Peterson, A., Difference Equations. An Introduction with Applications, 2nd Edition, Academic Press, New York, 2000. Kuczma, M., An Introduction to the Theory of Functional Equations and Inequalities, Polish Scientific Pub- lishers, Warsaw, 1985. Kuczma, M., Functional Equations in a Single Variable, Polish Scientific Publishers, Warsaw, 1968. Kuczma, M., Choczewski, B., and Ger, R., Iterative Functional Equations, Cambridge University Press, Cambridge, 1990. Mickens, R., Difference Equations, 2nd Edition, CRC Press, Boca Raton, 1991. Mirolyubov, A. A. and Soldatov, M. A., Linear Homogeneous Difference Equations [in Russian], Nauka Publishers, Moscow, 1981. Mirolyubov,A.A.andSoldatov,M.A., Linear Nonhomogeneous Difference Equations [in Russian], Nauka Publishers, Moscow, 1986. Nechepurenko, M. I., Iterations of Real Functions and Functional Equations [in Russian], Institute of Com- putational Mathematics and Mathematical Geophysics, Novosibirsk, 1997. 936 DIFFERENCE EQUATIONS AND OTHER FUNCTIONAL EQUATIONS Pelyukh, G. P. and Sharkovskii, O. M., Introduction to the Theory of Functional Equations [in Russian], Naukova Dumka, Kiev, 1974. Polyanin, A. D., Functional Equations, From Website EqWorld — The World of Mathematical Equations, http://eqworld.ipmnet.ru/en/solutions/fe.htm. Polyanin, A. D. and Manzhirov, A. V., Handbook of Integral Equations: Exact Solutions (Supplement. Some Functional Equations) [in Russian], Faktorial, Moscow, 1998. Polyanin, A. D. and Zaitsev, V. F., Handbook of Nonlinear Partial Differential Equations (Sections S.4 and S.5), Chapman & Hall/CRC Press, Boca Raton, 2004. Polyanin, A. D., Zaitsev, V. F., and Zhurov, A. I., Methods for the Solution of Nonlinear Equations of Mathematical Physics and Mechanics (Chapter 16. Solution of Some Functional Equations) [in Russian], Fizmatlit, Moscow, 2005. Rassias, T. M., Functional Equations and Inequalities, Kluwer Academic, Dordrecht, 2000. Samarskii, A. A. and Karamzin, Yu. N., Difference Equations [in Russian], Nauka Publishers, Moscow, 1978. Samarskii, A. A. and Nikolaev, E. S., Methods for the Solution of Grid Equations [in Russian], Nauka Publishers, Moscow, 1972. Sedaghat, H., Nonlinear Difference Equations Theory with Applications to Social Science Models,Kluwer Academic, Dordrecht, 2003. Sharkovsky,A.N.,Maistrenko,Yu.L.,andRomanenko,E.Yu.,Difference Equations and Their Applica- tions, Kluwer Academic, Dordrecht, 1993. Small, C. G., Functional Equations and How to Solve Them, Springer-Verlag, Berlin, 2007. Smital,J.andDravecky,J.,On Functions and Functional Equations, Adam Hilger, Bristol-Philadelphia, 1988. Zwillinger, D., CRC Standard Mathematical Tables and Formulae, 31st Edition, Chapman & Hall/CRC Press, Boca Raton, 2003. Chapter 18 Special Functions and Their Properties  Throughout Chapter 18 it is assumed that n is a positive integer unless otherwise specified. 18.1. Some Coefficients, Symbols, and Numbers 18.1.1. Binomial Coefficients 18.1.1-1. Definitions. C k n =  n k  = n! k!(n – k)! ,wherek = 1, , n; C 0 a = 1, C k a =  a k  =(–1) k (–a) k k! = a(a – 1) (a – k + 1) k! ,wherek = 1, 2, Here a is an arbitrary real number. 18.1.1-2. Generalization. Some properties. General case: C b a = Γ(a + 1) Γ(b + 1)Γ(a – b + 1) ,whereΓ(x) is the gamma function. Properties: C 0 a = 1, C k n = 0 for k =–1,–2, or k > n, C b+1 a = a b + 1 C b a–1 = a – b b + 1 C b a , C b a + C b+1 a = C b+1 a+1 , C n –1/2 = (–1) n 2 2n C n 2n =(–1) n (2n – 1)!! (2n)!! , C n 1/2 = (–1) n–1 n2 2n–1 C n–1 2n–2 = (–1) n–1 n (2n – 3)!! (2n – 2)!! , C 2n+1 n+1/2 =(–1) n 2 –4n–1 C n 2n , C n 2n+1/2 = 2 –2n C 2n 4n+1 , C 1/2 n = 2 2n+1 πC n 2n , C n/2 n = 2 2n π C (n–1)/2 n . 937 938 SPECIAL FUNCTIONS AND THEIR PROPERTIES 18.1.2. Pochhammer Symbol 18.1.2-1. Definition. (a) n = a(a + 1) (a + n – 1)= Γ(a + n) Γ(a) =(–1) n Γ(1 – a) Γ(1 – a – n) . 18.1.2-2. Some properties (k = 1, 2, ). (a) 0 = 1,(a) n+k =(a) n (a + n) k ,(n) k = (n + k – 1)! (n – 1)! , (a) –n = Γ(a – n) Γ(a) = (–1) n (1 – a) n ,wherea ≠ 1, , n; (1) n = n!, (1/2) n = 2 –2n (2n)! n! ,(3/2) n = 2 –2n (2n + 1)! n! , (a + mk) nk = (a) mk+nk (a) mk ,(a + n) n = (a) 2n (a) n ,(a + n) k = (a) k (a + k) n (a) n . 18.1.3. Bernoulli Numbers 18.1.3-1. Definition. The Bernoulli numbers are defined by the recurrence relation B 0 = 1, n–1  k=0 C k n B k = 0, n = 2, 3, Numerical values: B 0 = 1, B 1 =– 1 2 , B 2 = 1 6 , B 4 =– 1 30 , B 6 = 1 42 , B 8 =– 1 30 , B 10 = 5 66 , , B 2m+1 = 0 for m = 1, 2, All odd-numbered Bernoulli numbers but B 1 are zero; alleven-numbered Bernoulli numbers have alternating signs. The Bernoulli numbers are the values of Bernoulli polynomials at x = 0: B n = B n (0). 18.1.3-2. Generating function. Generating function: x e x – 1 = ∞  n=0 B n x n n! , |x| < 2π. This relation may be regarded as a definition of the Bernoulli numbers. The following expansions may be used to calculate the Bernoulli numbers: tan x = ∞  n=1 |B 2n | 2 2n (2 2n – 1) (2n)! x 2n , |x| < π 2 ; cot x = ∞  n=0 (–1) n B 2n 2 2n (2n)! x 2n–1 , |x| < π. 18.2. ERROR FUNCTIONS.EXPONENTIAL AND LOGARITHMIC INTEGRALS 939 18.1.4. Euler Numbers 18.1.4-1. Definition. The Euler numbers E n are defined by the recurrence relation n  k=0 C 2k 2n E 2k = 0 (even numbered), E 2n+1 = 0 (odd numbered), where n = 0, 1, Numerical values: E 0 = 1, E 2 =–1, E 4 = 5, E 6 =–61, E 8 = 1385, E 10 =–50251, , E 2n+1 = 0 for n = 0, 1, All Euler numbers are integer, the odd-numbered Euler numbers are zero, and the even- numbered Euler numbers have alternating signs. The Euler numbers are expressed via the values of Euler polynomials at x = 1/2: E n = 2 n E n (1/2), where n = 0, 1, 18.1.4-2. Generating function. Integral representation. Generating function: e x e 2x + 1 = ∞  n=0 E n x n n! , |x| < 2π. This relation may be regarded as a definition of the Euler numbers. Representation via a definite integral: E 2n =(–1) n 2 2n+1  ∞ 0 t 2n dt cosh(πt) . 18.2. Error Functions. Exponential and Logarithmic Integrals 18.2.1. Error Function and Complementary Error Function 18.2.1-1. Integral representations. Definitions: erf x = 2 √ π  x 0 exp(–t 2 ) dt (error function, also called probability integral), erfc x = 1 –erfx = 2 √ π  ∞ x exp(–t 2 ) dt (complementary error function). Properties: erf(–x)=–erfx;erf(0)=0,erf(∞)=1; erfc(0)=1, erfc(∞)=0. 940 SPECIAL FUNCTIONS AND THEIR PROPERTIES 18.2.1-2. Expansions as x → 0 and x →∞.Definite integral. Expansion of erf x into series in powers of x as x → 0: erf x = 2 √ π ∞  k=0 (–1) k x 2k+1 k!(2k + 1) = 2 √ π exp  –x 2  ∞  k=0 2 k x 2k+1 (2k + 1)!! . Asymptotic expansion of erfc x as x →∞: erfc x = 1 √ π exp  –x 2   M–1  m=0 (–1) m  1 2  m x 2m+1 + O  |x| –2M–1   , M = 1, 2, Integral:  x 0 erf tdt = x erf x – 1 2 + 1 2 exp(–x 2 ). 18.2.2. Exponential Integral 18.2.2-1. Integral representations. Definition: Ei(x)=  x –∞ e t t dt =–  ∞ –x e –t t dt for x < 0, Ei(x) = lim ε→+0   –ε –∞ e t t dt +  x ε e t t dt  for x > 0. Other integral representations: Ei(–x)=–e –x  ∞ 0 x sin t + t cos t x 2 + t 2 dt for x > 0, Ei(–x)=e –x  ∞ 0 x sin t – t cos t x 2 + t 2 dt for x < 0, Ei(–x)=–x  ∞ 1 e –xt ln tdt for x > 0, Ei(x)=C +lnx +  x 0 e t – 1 t dt for x > 0, where C = 0.5772 is the Euler constant. 18.2.2-2. Expansions as x → 0 and x →∞. Expansion into series in powers of x as x → 0: Ei(x)= ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ C +ln(–x)+ ∞  k=1 x k k! k if x < 0, C +lnx + ∞  k=1 x k k! k if x > 0. Asymptotic expansion as x →∞: Ei(–x)=e –x n  k=1 (–1) k (k – 1)! x k + R n , R n < n! x n . . bilinear functional equation of the form (17.5.5.8). Using formulas (17.5.5.9) for the construction of its solution, one can first obtain a system of ODEs and then find solutions of the original equation. D. and Manzhirov, A. V., Handbook of Integral Equations: Exact Solutions (Supplement. Some Functional Equations) [in Russian], Faktorial, Moscow, 1998. Polyanin, A. D. and Zaitsev, V. F., Handbook. k 2 ,thetermse k 2 ϕ and e –k 2 z in (17.5.5.22) should be replaced with ϕe k 1 ϕ and ze –k 1 z , respectively. In the case of purely imaginary or complex roots, one should separate the real (or imaginary) part of