Differences of fractional order
Authors:
J. B. Díaz and T. J. Osler
Journal:
Math. Comp. 28 (1974), 185202
MSC:
Primary 39A05; Secondary 26A33
MathSciNet review:
0346352
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Abstract: Derivatives of fractional order, , have been considered extensively in the literature. However, little attention seems to have been given to finite differences of fractional order, . In this paper, a definition of differences of arbitrary order is presented, and is computed for several specific functions f (Table 2.1). We find that the operator is closely related to the contour integral which defines Meijer's Gfunction. A Leibniz rule for the fractional difference of the product of two functions is discovered and used to generate series expansions involving the special functions.
 [1]
G. T. Cargo & O. Shisha, "Zeros of polynomials and fractional order differences of their coefficients," J. Math. Anal. Appl., v. 7, 1963, pp. 176182. MR 28 #2195. MR 0158974 (28:2195)
 [2]
E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable, Oxford Univ. Press, Oxford, 1935.
 [3]
H. T. Davis, The Summation of Series, Principia Press of Trinity Univ., San Antonio, Texas, 1962. MR 25 #5305. MR 0141908 (25:5305)
 [4]
B. Kuttner, "On differences of fractional order," Proc. London Math. Soc., (3), v. 7, 1957, pp. 453466. MR 20 #1131. MR 0094618 (20:1131)
 [5]
Y. L. Luke, The Special Functions and Their Approximations. Vols. I, II, Math. in Sci. and Engineering, vol. 53, Academic Press, New York, 1969. MR 39 #3039; MR 40 #2909.
 [6]
T. J. Osler, "Leibniz rule for fractional derivatives generalized and an application to infinite series," SIAM J. Appl. Math., v. 18, 1970, pp. 658674. MR 0260942 (41:5562)
 [7]
T. J. Osler, "The fractional derivative of a composite function," SIAM J. Math Anal., v. 1, 1970, pp. 288293. MR 41 #5563. MR 0260943 (41:5563)
 [8]
T. J. Osler, "Taylor's series generalized for fractional derivatives and applications," SIAM J. Math. Anal., v. 2, 1971, pp. 3748. MR 45 #3682. MR 0294612 (45:3682)
 [9]
T. J. Osler, "Fractional derivatives and Leibniz rule," Amer. Math. Monthly, v. 78, 1971, pp. 645649. MR 1536368
 [10]
T. J. Osler, "A further extension of the Leibniz rule to fractional derivatives and its relation to Parseval's formula," SIAM J. Math. Anal., v. 3, 1972, pp. 116. MR 0323970 (48:2323a)
 [11]
T. J. Osler, "An integral analogue of Taylor's series and its use in computing Fourier transforms," Math. Comp., v. 26, 1972, pp. 449460. MR 0306828 (46:5950)
 [12]
T. J. Osler, "A correction to Leibniz rule for fractional derivatives," SIAM J. Math. Anal., v. 4, 1973, pp. 456459. MR 0323971 (48:2323b)
 [13]
T. J. Osler, "The integral analog of the Leibniz rule," Math. Comp., v. 26, 1972, pp. 903915. MR 0314240 (47:2792)
 [14]
I. N. Sneddon, Fourier Transforms, McGrawHill, New York, 1951. MR 13, 29. MR 0041963 (13:29h)
 [15]
E. C. Titchmarsh, The Theory of Functions, 2nd ed., Oxford Univ. Press, Oxford, 1939.
 [1]
 G. T. Cargo & O. Shisha, "Zeros of polynomials and fractional order differences of their coefficients," J. Math. Anal. Appl., v. 7, 1963, pp. 176182. MR 28 #2195. MR 0158974 (28:2195)
 [2]
 E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable, Oxford Univ. Press, Oxford, 1935.
 [3]
 H. T. Davis, The Summation of Series, Principia Press of Trinity Univ., San Antonio, Texas, 1962. MR 25 #5305. MR 0141908 (25:5305)
 [4]
 B. Kuttner, "On differences of fractional order," Proc. London Math. Soc., (3), v. 7, 1957, pp. 453466. MR 20 #1131. MR 0094618 (20:1131)
 [5]
 Y. L. Luke, The Special Functions and Their Approximations. Vols. I, II, Math. in Sci. and Engineering, vol. 53, Academic Press, New York, 1969. MR 39 #3039; MR 40 #2909.
 [6]
 T. J. Osler, "Leibniz rule for fractional derivatives generalized and an application to infinite series," SIAM J. Appl. Math., v. 18, 1970, pp. 658674. MR 0260942 (41:5562)
 [7]
 T. J. Osler, "The fractional derivative of a composite function," SIAM J. Math Anal., v. 1, 1970, pp. 288293. MR 41 #5563. MR 0260943 (41:5563)
 [8]
 T. J. Osler, "Taylor's series generalized for fractional derivatives and applications," SIAM J. Math. Anal., v. 2, 1971, pp. 3748. MR 45 #3682. MR 0294612 (45:3682)
 [9]
 T. J. Osler, "Fractional derivatives and Leibniz rule," Amer. Math. Monthly, v. 78, 1971, pp. 645649. MR 1536368
 [10]
 T. J. Osler, "A further extension of the Leibniz rule to fractional derivatives and its relation to Parseval's formula," SIAM J. Math. Anal., v. 3, 1972, pp. 116. MR 0323970 (48:2323a)
 [11]
 T. J. Osler, "An integral analogue of Taylor's series and its use in computing Fourier transforms," Math. Comp., v. 26, 1972, pp. 449460. MR 0306828 (46:5950)
 [12]
 T. J. Osler, "A correction to Leibniz rule for fractional derivatives," SIAM J. Math. Anal., v. 4, 1973, pp. 456459. MR 0323971 (48:2323b)
 [13]
 T. J. Osler, "The integral analog of the Leibniz rule," Math. Comp., v. 26, 1972, pp. 903915. MR 0314240 (47:2792)
 [14]
 I. N. Sneddon, Fourier Transforms, McGrawHill, New York, 1951. MR 13, 29. MR 0041963 (13:29h)
 [15]
 E. C. Titchmarsh, The Theory of Functions, 2nd ed., Oxford Univ. Press, Oxford, 1939.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197403463525
PII:
S 00255718(1974)03463525
Keywords:
Finite difference,
fractional differences,
fractional derivatives,
Leibniz rule,
special functions,
MellinBarnes contour integrals
Article copyright:
© Copyright 1974
American Mathematical Society
