## Fq-linear calculus over function fields (1999)

Venue: | J. Number Theory |

Citations: | 8 - 8 self |

### BibTeX

@ARTICLE{Kochubei99fq-linearcalculus,

author = {Anatoly N. Kochubei},

title = {Fq-linear calculus over function fields},

journal = {J. Number Theory},

year = {1999},

pages = {281--300}

}

### OpenURL

### Abstract

We define analogues of higher derivatives for Fq-linear functions over the field of formal Laurent series with coefficients in Fq. This results in a formula for Taylor coefficients of a Fq-linear holomorphic function, a definition of classes of Fq-linear smooth functions which are characterized in terms of coefficients of their Fourier-Carlitz expansions. A Volkenborn-type integration theory for Fq-linear functions is developed; in particular, an integral representation of the Carlitz logarithm is obtained. Key words: Fq-linear function; Carlitz basis; Carlitz logarithm; Volkenborn integral; difference operator; Bargmann-Fock representation.

### Citations

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Finite Fields
- Lidl, Niederreiter
- 1984
(Show Context)
Citation Context ...q d −1 |cn| → 0 for n → ∞ instead of (18). The proof is based on the fact that the absolute value |[i]|π equals q −1 if d divides i, or 1 in the opposite case (which can be deduced from Lemma 2.13 in =-=[13]-=-). 6 INDEFINITE SUM Viewing the operator a − as a kind of a derivative, it is natural to introduce an appropriate antiderivative. Following the terminology used in the analysis over Zp (see [15]) we c... |

391 |
Generalized Coherent States and their Applications
- Perelomov
- 1986
(Show Context)
Citation Context ...s based on identities for the difference operator ∆ = ∆ (1) , which emerge as a function field analogue of the Bargmann-Fock representation of the canonical commutation relations of quantum mechanics =-=[9, 14]-=-. Note that an analogue of the Schrödinger representation was obtained in [12]. We will also use ∆ (n) in order to characterize “smoothness” of Fq-linear functions and obtain an exact correspondence b... |

150 |
Basic Structures of Function Field Arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete (3
- Goss
- 1996
(Show Context)
Citation Context ...f(gt) dt = g f(t) dt. O Our last result will contain the calculation of integrals for some important functions on O. Let us recall the definitions of some special functions introduced by Carlitz (see =-=[8]-=-). The function Cs(z), s ∈ Fq[x], z ∈ K, defining the Carlitz module, is given by the formula i=0 O deg s ∑ Cs(z) = fi(s)z qi . If |z| < 1, Cs(z) can be extended with respect to s: Cs(z) = ∞∑ i=0 fi(s... |

62 |
The Segal-Bargmann ”Coherent State” transform for compact Lie groups
- Hall
- 1994
(Show Context)
Citation Context ...s based on identities for the difference operator ∆ = ∆ (1) , which emerge as a function field analogue of the Bargmann-Fock representation of the canonical commutation relations of quantum mechanics =-=[9, 14]-=-. Note that an analogue of the Schrödinger representation was obtained in [12]. We will also use ∆ (n) in order to characterize “smoothness” of Fq-linear functions and obtain an exact correspondence b... |

49 |
On certain functions connected with polynomials in a Galois field
- Carlitz
- 1935
(Show Context)
Citation Context ...h leads, in particular, to an integral representation of the Carlitz logarithm, establishing a direct connection between the latter and the Carlitz module operation. 2 PRELIMINARIES 2.1 Carlitz Basis =-=[3, 4, 6, 12, 20]-=- Denote by | · | the non-Archimedean absolute value on K; if z ∈ K, z = ∞∑ ζix i , n ∈ Z, ζi ∈ Fq , ζn ̸= 0, i=n 3then |z| = q −n . It is well known that this valuation can be extended onto Kc. Let O... |

44 |
Ultrametric calculus
- Schikhof
- 1984
(Show Context)
Citation Context ...nalyticity) and the decay rate for coefficients of Fourier-Carlitz expansions. Note that in the p-adic case (where smoothness is understood in a conventional sense) such a correspondence was found in =-=[1, 2, 15]-=-. Having a new “derivative”, we can introduce a kind of an antiderivative, and a Volkenborn type integral (see [15, 19] for the case of zero characteristic), which leads, in particular, to an integral... |

22 |
nombres p-adiques
- Les
- 1975
(Show Context)
Citation Context ...nalyticity) and the decay rate for coefficients of Fourier-Carlitz expansions. Note that in the p-adic case (where smoothness is understood in a conventional sense) such a correspondence was found in =-=[1, 2, 15]-=-. Having a new “derivative”, we can introduce a kind of an antiderivative, and a Volkenborn type integral (see [15, 19] for the case of zero characteristic), which leads, in particular, to an integral... |

20 |
A set of polynomials
- Carlitz
- 1940
(Show Context)
Citation Context ...h leads, in particular, to an integral representation of the Carlitz logarithm, establishing a direct connection between the latter and the Carlitz module operation. 2 PRELIMINARIES 2.1 Carlitz Basis =-=[3, 4, 6, 12, 20]-=- Denote by | · | the non-Archimedean absolute value on K; if z ∈ K, z = ∞∑ ζix i , n ∈ Z, ζi ∈ Fq , ζn ̸= 0, i=n 3then |z| = q −n . It is well known that this valuation can be extended onto Kc. Let O... |

13 |
Hypergeometric functions for function fields
- Thakur
(Show Context)
Citation Context ...ficients from the Galois field Fq, q = p γ , γ ∈ Z+. The foundations of analysis over K were laid in a series of papers by Carlitz (see, in particular, [3-5]), Wagner [20, 21], Goss [6-8], and Thakur =-=[17, 18]-=-. An interesting property of many functions introduced in the above works as analogues of classical elementary and special functions, is their Fq-linearity. Recall that a function f : K → Kc (where Kc... |

12 |
Fourier series, measures and divided power series in the theory of function fields, K-Theory 1
- Goss
- 1989
(Show Context)
Citation Context ...h leads, in particular, to an integral representation of the Carlitz logarithm, establishing a direct connection between the latter and the Carlitz module operation. 2 PRELIMINARIES 2.1 Carlitz Basis =-=[3, 4, 6, 12, 20]-=- Denote by | · | the non-Archimedean absolute value on K; if z ∈ K, z = ∞∑ ζix i , n ∈ Z, ζi ∈ Fq , ζn ̸= 0, i=n 3then |z| = q −n . It is well known that this valuation can be extended onto Kc. Let O... |

10 |
Linear operators in local fields of prime characteristic
- WAGNER
(Show Context)
Citation Context ...f formal Laurent series with coefficients from the Galois field Fq, q = p γ , γ ∈ Z+. The foundations of analysis over K were laid in a series of papers by Carlitz (see, in particular, [3-5]), Wagner =-=[20, 21]-=-, Goss [6-8], and Thakur [17, 18]. An interesting property of many functions introduced in the above works as analogues of classical elementary and special functions, is their Fq-linearity. Recall tha... |

9 | Harmonic oscillator in characteristic p
- Kochubei
- 1998
(Show Context)
Citation Context ...unction field analogue of the Bargmann-Fock representation of the canonical commutation relations of quantum mechanics [9, 14]. Note that an analogue of the Schrödinger representation was obtained in =-=[12]-=-. We will also use ∆ (n) in order to characterize “smoothness” of Fq-linear functions and obtain an exact correspondence between the degree of smoothness (or analyticity) and the decay rate for coeffi... |

7 | Some special functions over GF(q, x - Carlitz - 1960 |

7 |
Interpolation series for continuous functions on π-adic completions of GF(q, x
- Wagner
- 1971
(Show Context)
Citation Context ...f formal Laurent series with coefficients from the Galois field Fq, q = p γ , γ ∈ Z+. The foundations of analysis over K were laid in a series of papers by Carlitz (see, in particular, [3-5]), Wagner =-=[20, 21]-=-, Goss [6-8], and Thakur [17, 18]. An interesting property of many functions introduced in the above works as analogues of classical elementary and special functions, is their Fq-linearity. Recall tha... |

6 | p-adic commutation relations
- Kochubei
- 1996
(Show Context)
Citation Context ...of the Hilbert spaces of square integrable functions where the operators of conventional quantum mechanics are defined. Note also that a p-adic analogue of the Schrödinger representation was found in =-=[11]-=-. It is interesting that the above operators possess remarkable algebraic properties. The operator ∆ is a derivation on the ring of Fq-linear functions from O to itself, with composition being the mul... |

5 |
Quantum Mechanics and the Particles of Nature, Cambridge
- Sudbery
- 1986
(Show Context)
Citation Context ...to the orthonormality of {hj}. Since σm,q m −1 = g0(0) 2.2 Canonical Commutation Relations [12] Γ0 Γq m −1−j λkσk,j, . = 1, we come to (3). ✷ In the quantum mechanics of harmonic oscillator (see e.g. =-=[16]-=-) a creation operator transforms a stationary state into a stationary state of the next (higher) energy level, an annihilation operator acts in the opposite way. In quantum field theory these properti... |

2 | A formal Mellin transform in the arithmetic of function fields - Goss - 1991 |

2 |
Galois Calculus and Carlitz Exponentials. In: The Arithmetic of Function Fields, (eds: D. Goss et al), de Gruyter
- Hellegouarch
- 1992
(Show Context)
Citation Context ...ies. The operator ∆ is a derivation on the ring of Fq-linear functions from O to itself, with composition being the multiplication in the ring. The operator Rq − I is an Rq-derivation in the sense of =-=[10]-=-. The Bargmann-Fock representation is a realization of a structure like (4)-(6) by operators on a space of holomorphic functions. A simple construction for the case of a function field is given below.... |

1 |
Fonctions k-lipschitziennes sur anneau local et polynômes à valeurs entières
- Barsky
- 1973
(Show Context)
Citation Context ...nalyticity) and the decay rate for coefficients of Fourier-Carlitz expansions. Note that in the p-adic case (where smoothness is understood in a conventional sense) such a correspondence was found in =-=[1, 2, 15]-=-. Having a new “derivative”, we can introduce a kind of an antiderivative, and a Volkenborn type integral (see [15, 19] for the case of zero characteristic), which leads, in particular, to an integral... |

1 |
Ein p-adisches Integral und seine Anwendungen
- Volkenborn
- 1972
(Show Context)
Citation Context ...ness is understood in a conventional sense) such a correspondence was found in [1, 2, 15]. Having a new “derivative”, we can introduce a kind of an antiderivative, and a Volkenborn type integral (see =-=[15, 19]-=- for the case of zero characteristic), which leads, in particular, to an integral representation of the Carlitz logarithm, establishing a direct connection between the latter and the Carlitz module op... |