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G-Transform

The G-transform of a function f(x) is defined by the integral

(Gf)(x)=(G_(pq)^(mn)|(a_p); (b_q)|f(t))(x)
(1)
=1/(2pii)int_sigmaGamma[(b_m)+s, 1-(a_n)-s; (a_p^(n+1))+s, 1-(b_q^(m+1))-s]×f^*(s)x^(-s)ds,
(2)

where G_(pq)^(mn) is the Meijer G-function,

Gamma[(b_m)+s, 1-(a_n)-s; (a_p^(n+1))+s, 1-(b_q^(m+1))-s]
(3)
=Gamma[b_1+s, ..., b_m+s, 1-a_1-s, ..., 1-a_n-s; a_(n+1)+s, ..., a_p+s, 1-b_(m+1)-s, ..., 1-b_q-s]
(4)
=(product_(j=1)^(m)Gamma(b_j+s)product_(j=1)^(n)Gamma(1-a_j-s))/(product_(j=n+1)^(p)Gamma(a_j+s)product_(j=m+1)^(q)Gamma(1-b_j-s)),
(5)

f^*(s) is the Mellin transform of a function f(x), sigma is the contour sigma={1/2-iinfty,1/2+iinfty}, (a_n)=a_1,a_2,...,a_n, (a_p^(n+1))=a_(n+1),a_(n+2),...,a_p, (b_m)=b_1,...,b_m, (b_q^(m+1))=b_(m+1),...,b_q, and the components of the vectors (a_p) and (b_q) are complex numbers satisfying the conditions R[a_p])!=1/2, 3/2, 5/2, ... and R[b_q]!=-1/2, -3/2, -5/2, ....

See also

Meijer G-Function, W-Transform

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References

Samko, S. G.; Kilbas, A. A.; and Marichev, O. I. "Definition of the G-Transform. The Spaces M_(c,gamma)^(-1)(L) and L_2^((c,gamma)) and Their Characterization." §36.1 in Fractional Integrals and Derivatives. Yverdon, Switzerland: Gordon and Breach, pp. 704-709, 1993.

Referenced on Wolfram|Alpha

G-Transform

Cite this as:

Weisstein, Eric W. "G-Transform." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/G-Transform.html

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