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The polynomials M_k(x;delta,eta) which form the Sheffer sequence for g(t) = {[1+deltaf(t)]^2+[f(t)]^2}^(eta/2) (1) f(t) = tan(t/(1+deltat)) (2) which have generating function ...

An integral equation of the form f(x)=int_a^xK(x,t)phi(t)dt, where K(x,t) is the integral kernel, f(x) is a specified function, and phi(t) is the function to be solved for.

An integral equation of the form phi(x)=f(x)+int_a^xK(x,t)phi(t)dt, where K(x,t) is the integral kernel, f(x) is a specified function, and phi(t) is the function to be solved ...

For R[n]>-1 and R[z]>0, Pi(z,n) = n^zint_0^1(1-x)^nx^(z-1)dx (1) = (n!)/((z)_(n+1))n^z (2) = B(z,n+1), (3) where (z)_n is the Pochhammer symbol and B(p,q) is the beta ...

Q_n^((alpha,beta))(x)=2^(-n-1)(x-1)^(-alpha)(x+1)^(-beta) ×int_(-1)^1(1-t)^(n+alpha)(1+t)^(n+beta)(x-t)^(-n-1)dt. In the exceptional case n=0, alpha+beta+1=0, a nonconstant ...

The cylinder function is defined as C(x,y)={1 for sqrt(x^2+y^2)<=a; 0 for sqrt(x^2+y^2)>a. (1) The Bessel functions are sometimes also called cylinder functions. To find the ...

For r and x real, with 0<=arg(sqrt(k^2-tau^2))<pi and 0<=argk<pi, 1/2iint_(-infty)^inftyH_0^((1))(rsqrt(k^2-tau^2))e^(itaux)dtau=(e^(iksqrt(r^2+x^2)))/(sqrt(r^2+x^2)), where ...

Let H_nu^((iota))(x) be a Hankel function of the first or second kind, let x,nu>0, and define w=sqrt((x/nu)^2-1). Then ...

The Legendre polynomials, sometimes called Legendre functions of the first kind, Legendre coefficients, or zonal harmonics (Whittaker and Watson 1990, p. 302), are solutions ...

Debye's asymptotic representation is an asymptotic expansion for a Hankel function of the first kind with nu approx x. For 1-nu/x>epsilon, nu/x=sinalpha, ...