References
121
toc
erg = reshape(erg,size(c));
% visualization
surface(Cr,Ci,abs(erg),angle(erg)), zlim([0,5])
colorbar, caxis([-pi,pi]), shg
shading interp, colormap(jet);
ylabel(’imag(c)’); xlabel(’real(c)’);
view(24.5, 24.5), light
Many orthogonal polynomials can be directly evaluated via hypergeometric
functions. In the following we will list a few “single line” [4] examples:
The Krawtchonk polynomials can be evaluated via
K n (x; p, N) = 2 F 1 (−n, −x; −N; 1./p) with
(8.20)
n ∈ N, n ≤ N ∈ N and 0 < p < 1. Thus,
>> objK = gausshyp(-n,-x,-N,1./p), leads to a finite polynomial.
The Meixner polynomials are given by
M n (x; b, p) = 2 F 1 (−n, −x; b; 1 − 1/p) with b > 0
(8.21)
and hence >> objM = gausshyp(-n,-x,b,1-1./p).
The Meixner–Pollaczek polynomial can be evaluated via
P
λ
n (x; Φ) =
(2λ) n
n!
exp(inΦ) 2 F 1 (−n, λ + i · x; 2λ; 1 − exp(−2iΦ)) with
(8.22)
λ > 0, 0 ≤ Φ ≤ π and hence
objMP=gausshyp(-n,lambda+i * x,2 * lambda,1-exp(-2 * i * Phi));
polvalue=pochC(2 * lambda,n).value/gammaC(n+1).value ...
. * exp(i * n * Phi) * objMP.value;
The first line creates an object of the class gausshyp. Its property value will be
multiplied with prefactor to obtain the final polynomial value. Using the class
pochC and the class gammaC allows to generalize this polynomial into the
complex domain.
References
1. Flügge, S.: Practical Quantum Mechanics. Springer, Berlin (1994)
2. Gil, A., Segura, J., Temme, N.M.: Numerically satisfactory solutions of hypergeometric
recursions. Report MAS-R0608 (2006)
3. Gradstein, I.S., Rhysik, I.M.: Tafeln ·Tables II. Verlag Harry Deutsch Thun, Frankfurt A. M.
(1981)
4. Olver, F.W.J., Olde Daalhuis, A.B., Lozier, D.W., Schneider, B.I., Boisvert, R.F., Clark, C.W.,
Miller, B.R., Sounders, B.V. (eds.): NIST Digital Library of Mathematical Functions. (2017).
http://dlmf.nist.gov. Rel. 1.0.17
121
toc
erg = reshape(erg,size(c));
% visualization
surface(Cr,Ci,abs(erg),angle(erg)), zlim([0,5])
colorbar, caxis([-pi,pi]), shg
shading interp, colormap(jet);
ylabel(’imag(c)’); xlabel(’real(c)’);
view(24.5, 24.5), light
Many orthogonal polynomials can be directly evaluated via hypergeometric
functions. In the following we will list a few “single line” [4] examples:
The Krawtchonk polynomials can be evaluated via
K n (x; p, N) = 2 F 1 (−n, −x; −N; 1./p) with
(8.20)
n ∈ N, n ≤ N ∈ N and 0 < p < 1. Thus,
>> objK = gausshyp(-n,-x,-N,1./p), leads to a finite polynomial.
The Meixner polynomials are given by
M n (x; b, p) = 2 F 1 (−n, −x; b; 1 − 1/p) with b > 0
(8.21)
and hence >> objM = gausshyp(-n,-x,b,1-1./p).
The Meixner–Pollaczek polynomial can be evaluated via
P
λ
n (x; Φ) =
(2λ) n
n!
exp(inΦ) 2 F 1 (−n, λ + i · x; 2λ; 1 − exp(−2iΦ)) with
(8.22)
λ > 0, 0 ≤ Φ ≤ π and hence
objMP=gausshyp(-n,lambda+i * x,2 * lambda,1-exp(-2 * i * Phi));
polvalue=pochC(2 * lambda,n).value/gammaC(n+1).value ...
. * exp(i * n * Phi) * objMP.value;
The first line creates an object of the class gausshyp. Its property value will be
multiplied with prefactor to obtain the final polynomial value. Using the class
pochC and the class gammaC allows to generalize this polynomial into the
complex domain.
References
1. Flügge, S.: Practical Quantum Mechanics. Springer, Berlin (1994)
2. Gil, A., Segura, J., Temme, N.M.: Numerically satisfactory solutions of hypergeometric
recursions. Report MAS-R0608 (2006)
3. Gradstein, I.S., Rhysik, I.M.: Tafeln ·Tables II. Verlag Harry Deutsch Thun, Frankfurt A. M.
(1981)
4. Olver, F.W.J., Olde Daalhuis, A.B., Lozier, D.W., Schneider, B.I., Boisvert, R.F., Clark, C.W.,
Miller, B.R., Sounders, B.V. (eds.): NIST Digital Library of Mathematical Functions. (2017).
http://dlmf.nist.gov. Rel. 1.0.17