q -Fractional Calculus and Equations
This nine-chapter monograph introduces a rigorous investigation of q-difference operators in standard and fractional settings. It starts with elementary calculus of q-differences and integration of Jackson’s type before turning to q-difference equati
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Tables of Fractional Derivatives and q-Derivatives
In this appendix, we collect the Riemann–Liouville fractional derivative and Caputo fractional of some q-analogues of the celebrated special functions and we also include a table of Riemann–Liouville fractional derivative for comparison.
A.1 Table of Riemann–Liouville Fractional Derivatives Table A.1 Riemann–Liouville fractional derivatives
.x/ x ˇ1 e x
˛ D0C .x/; x > 0; ˛ > 0
.ˇ/ x ˇ˛1 ; ˇ > 0 .ˇ ˛/ .x/˛ E1;1˛ . x/
cos. x/
.ˇ/ x ˇ˛1 1 F1 .ˇI ˇ ˛I x/ .ˇ ˛/ x ˛ E1=2;1˛ . 2 x 2 /
sin. .x a//
x 1˛ E1=2;2˛ . 2 x 2 /
x ˇ1 E;ˇ . x /
x ˇ˛1 E;ˇ˛ . x /; ˇ; > 0
x ˇ1 2 F1 .; I ˇI x/
.ˇ/ x ˇ˛C1 2 F1 .; I ˇ ˛I x/ ; ˇ > 0 .ˇ ˛/
x ˇ1 e x
M.H. Annaby and Z.S. Mansour, q-Fractional Calculus and Equations, Lecture Notes in Mathematics 2056, DOI 10.1007/978-3-642-30898-7, © Springer-Verlag Berlin Heidelberg 2012
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A Tables of Fractional Derivatives and q-Derivatives
A.2 Table of Riemann–Liouville Fractional q-Derivatives Table A.2 Riemann–Liouville fractional q-derivatives
Dq˛
x ˇ1 ; ˇ > 0
q x ˇ˛1 q .ˇ˛/
eq . x/
x ˛ e1;1˛ . x.1 q/1 I q/
Eq . x/
x ˛ E1;1˛ . x.1 q/1 I q/
x ˇ1 eq . x/
x ˇ˛1 q .ˇ/ ˇ ˇ˛ ; q; x/; q .ˇ˛/ 2 1 .0; q I q
x ˇ1 Eq . x/
x ˇ˛1 q .ˇ/ .q ˇ I q ˇ˛ I q; x/; q .ˇ˛/ 1 1
cosq x
x ˛ e2;1˛ . 2 x 2 .1 q/2 I q/
sinq x
.1 q/1 x 1˛ e2;2˛ . 2 x 2 .1 q/2 I q/
Cosq x
x ˛ .q 2 I q 2˛ ; q 1˛ I q 2 ; q 2 x 2 / q .1˛/ 1 2
Sinq x
x 1˛ .q 2 I q 2˛ ; q 1˛ I q 2 ; q 3 2 x 2 / q .1˛/ 1 2
cos. xI q/
x ˛ E2;1˛ .q 2 x 2 I q/
sin. xI q/
x 1˛ E2;2˛ .q 2 2 x 2 I q/
x ˇ1 E;ˇ . x I q/
x ˇ˛1 E;ˇ˛ . x I q/; ˇ; > 0
x ˇ1 e;ˇ . x I q/
x ˇ˛1 e;ˇ˛ . x I q/; ˇ; > 0
x ˇ1 2 1 a; bI q ˇ I q; x
q .ˇ/x ˇ˛1 ˇ˛ I q; x ; ˇ > 0 2 1 a; bI q q .ˇ ˛/ q .ˇ/x ˇ˛1 ˇ ˇ˛ I q; x ; ˇ > 0 3 2 a; b; q I c; q q .ˇ ˛/
x ˇ1 2 1 .a; bI cI q; x/
x > 0;
˛>0
.ˇ/
ˇ>0
ˇ>0
A.3 Table of the Erd´eli–Kober Fractional q-Integral Operator The next table contains the Erd´eli–Kober fractional integrals for some q-functions. An extended table can be found in [271].
A.3 Table of the Erd´eli–Kober Fractional q-Integral Operator
297
;˛
Table A.3 The integral operator Iq ;˛ Iq .x > 0/ x ˇ1
x ˇ1
q . C ˇ/ ; Re .ˇ C / > 0 q . C ˇ C ˛/
x ˇ1 eq . x/;
x ˇ1
Cˇ CˇC˛ q . C ˇ/ Iq I q; x 2 1 0; q q . C ˇ C ˛/
x ˇ1
Cˇ C˛Cˇ q . C ˇ/ Iq I q; x 1 1 q q . C ˇ C ˛/
x ˇ1
q . C ˇ/ q . C ˇ C ˛/
Re .ˇ C / > 0 x ˇ1 Eq . x/; Re .ˇ C / > 0 x ˇ1 cosq x; Re .ˇ C / > 0 x ˇ1 Cosq . x/; Re .ˇ C / > 0 x ˇ1 cos. xI q/; Re .ˇ C / > 0 x ˇ1 sinq x; Re.ˇ C / > 1 x ˇ1 Sinq x; Re.ˇ C / > 1 x ˇ1 sin. xI q/ Re.ˇ C / > 1
4 3 .0; 0; q
Cˇ
; q CˇC1 I q ˇC˛C ; q ˇC˛CC1 I q 2 ; 2 x 2 /
x ˇ1 q . C ˇ/ q . C ˇ C ˛/ 2 3 .q
Cˇ
; q CˇC1 I q; q C˛Cˇ ; q C˛CˇC1 I q 2 ; q 2 x 2 /;
q .ˇ C / x ˇ1 q .ˇ C C ˛/
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