In te rn a tio n a l E le ctro n ic J o u rn a l o f P u re a n d A p p lie d M ...

International Electronic Journal of Pure and Applied Mathematics – IEJPAM, Volume 1, No. 3 (2010)
236 A. Anjorin
regular singular points at {0,1,a, ∞}, with the exponents of these singularities being
respectively, {0,1 −γ}, {0,1 −δ}, {0,1 −ǫ}, and {α,β}. The equation (1.1) is called
Heun’s equation (see [12]).
The hypergeometric equation (see [12])
z(1 − z) d2y + [c − (a + b + 1)z]dy − aby = 0, (1.2)
dz2 dz
has three regular singular points. In the works of [10], it has been shown that these
two equations above can be transformed to one another via six rational polynomials
z = R(t), where R(t) = t 2 ,1 − t 2 ,(t − 1) 2 ,2t − t 2 ,(2t − 1) 2 ,4t(1 − t). The following
parameter relations were deduced [10].
For the polynomial R(t) = t 2
• α + β = 2(a + b), αβ = 4ab, γ = −1 + 2c, δ = 1 + a + b − c, ǫ = δ, q = 0 and
d = −1.
For the polynomial R(t) = 1 − t 2
• α + β = 2(a + b), αβ = 4ab, γ = 1 − 2c + 2a + 2b, δ = c, ǫ = δ = c, q = 0 and
d = −1.
For the polynomial R(t) = (t − 1) 2
• α + β = 2(a + b), αβ = 4ab, γ = 1 + a + b − c, δ = −1 + 2c, ǫ = γ, q = 4ab
and d = 2.
For the polynomial R(t) = 2t − t 2
• α + β = 2(a + b), αβ = 4ab, γ = c, δ = 1 − 2c + 2a + 2b, ǫ = δ = c, q = 4ab
and d = 2.
For the polynomial R(t) = (2t − 1) 2
• α + β = 2(a + b), αβ = 4ab, γ = −1 + a + b − c, δ = γ, ǫ = δ = −1 + 2c,
q = 2ab and d = 1/2.
For the polynomial R(t) = 4t(1 − t)
• α + β = 2(a + b), αβ = 4ab, γ = c, δ = γ, ǫ = 1 − 2c + 2a + 2b, q = 2ab and
d = 1/2.
Assuming H(d,q;α,β,γ,δ,ǫ;t) and 2F1(a,b;c;z = R(t)) are representative forms
of the solutions of (1.1) and (1.2) respectively, toghether with the parametres above
relations can de established between these two forms via the polynomials data given
above. We provide an answer to this in the resent paper. Indeed, we provide that
the integral of the solution of GHE can be expressed in terms of another GHE
solution.