Titles and Short Summaries of the Talks

39 Functional Equations
Final: 2013/2/7
5 Katsuyuki Nishimoto
∗ Solutions to the homogeneous Bessel equation by means of N-fractional
(Descartes Press Co.) calculus operator · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 15
Summary: In this article, the solutions to the homogeneous Bessel equation are discussed by
means of the N-fractional calculus operator (NFCO). That is, we have the particular solutions as
follows, for example.
φ [1] = z ν e iz (e −i2z · z −(ν+1/2) ) ν−1/2 (fractional differintegrated form)
= (−i2) ν−1/2z −1/2e−iz 2F0( 1
2
( i
< 1)
2z
= A · H (2)
ν (z), (A = √ π2 ν−1 e −iπν )
and
1 i
− ν, 2 + ν; 2z )
φ [6] = z −ν e iz (z ν−1/2 · e −i2z ) −(ν+1/2) (fractional differintegrated form)
= eiπ(ν+1/2) Γ(−2ν)zν e−iz 1F1( 1
( )
2 + ν; 1 + 2ν; 2iz) (|2iz| < 1)
Γ(−2ν−k)
< ∞
Γ(−ν+1/2)
= B ∗ · J (2)
ν (z), (B ∗ = 2 ν Γ(−2ν)Γ(1 + ν)e iπ(ν+1/2) )
where pFq(· · · · · · ) is the generalized Gauss hypergeometric function, H (2)
ν (z) is the Hankel function,
and
J (2)
ν (z) = e−iz (z/2) ν
1
Γ(1+ν) 1F1( 2 + ν; 1 + 2ν; 2iz) = Jν(z) (|2iz| < 1)
is the first kind Bessel Function.
6 Katsuyuki Nishimoto
∗ The solutions to the radial Schrödinger equation of the hydrogen atom
(Descartes Press Co.) by means of N-fractional calculus operator · · · · · · · · · · · · · · · · · · · · · · · · 15
Summary: In this article, the solutions to the radial Schrödinger equation of the Hydrogen atom
(in the Coulomb field)
φ2 · x 2 + φ1 · 2x + φ · {−(1/4)x 2 + νx + l(l + 1)} = 0
are discussed by means of N-fractional calculus operator.
A particular solution to the equation above is shown as follows for example.
φ = φ [1] = x l e x/2 (e −x · x ν−(l+1) )l+ν (fractional differintegrated form)
= (e iπ ) l+ν e −x/2 x ν−1 2F0(−l − ν, l + 1 − ν; −1/x)
(| − 1/x| < 1)
where pFq(· · · · · · ); Generalized Gauss hypergeometric functions.
7 Ryu Sasaki (Kyoto Univ.) ♯ Global solutions of certain second order differential equations with a
Kouichi Takemura (Chuo Univ.) high degree of apparent singularity · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 10
Summary: Infinitely many explicit solutions of certain second order differential equations with an
apparent singularity of characteristic exponent −2 are constructed by adjusting the parameter of
the multi-indexed Laguerre polynomials.
8 Nobuki Takayama
(Kobe Univ./JST CREST)
♯ Pfaffian systems of A-hypergeometric sysytems · · · · · · · · · · · · · · · · · · · · 15
Takayuki Hibi
(Osaka Univ./JST CREST)
Kenta Nishiyama
(Osaka Univ./JST CREST)
Summary: We are interested in bases of Rn/(RnHA[s]) as the vector space over the field C(s, x)
where HA[s] is an A-hypergeometric ideal. Any basis of the vector space yields an associated
Pfaffian system or an integrable connection associated to HA[s]. Bases can be described by those
of simpler quotients Rn/(inw(IA) + ∑ Rn(Ei − si)).