Two chain rules for divided differences and Faà di Bruno’s formula
نویسندگان
چکیده
منابع مشابه
Two chain rules for divided differences and Faà di Bruno's formula
In this paper we derive two formulas for divided differences of a function of a function. Both formulas lead to other divided difference formulas, such as reciprocal and quotient rules. The two formulas can also be used to derive Faà di Bruno’s formula and other formulas for higher derivatives of composite functions. We also derive a divided difference version of Faà di Bruno’s determinant form...
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In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: Keywords: Complete Bell polynomial Determinant Faà di Bruno's formula q-analogue a b s t r a c t Faà di Bruno's formula is t...
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where the sum is over all different solutions in nonnegative integers b1, . . . , bm of b1 + 2b2 + · · · + mbm = m, and k := b1 + · · · + bm. For example, when m = 3 this instructs us to look at all solutions in nonnegative integers of the equation b1 + 2b2 + 3b3 = 3. We can have b3 = 1, b1 = 0 = b2, in which case k = 1 and we get the term 3! 0! 0! 1! ′ ( f (t)) ( f ′′′(t) 3! ) = g′ ( f (t)) f ...
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ژورنال
عنوان ژورنال: Mathematics of Computation
سال: 2006
ISSN: 0025-5718,1088-6842
DOI: 10.1090/s0025-5718-06-01916-8