The -transform of is defined by

(1) |

(2) |

(3) |

(4) |

(5) |

(6) | |||

(7) | |||

(8) | |||

(9) | |||

(10) | |||

(11) | |||

(12) | |||

(13) | |||

(14) |

Transforms of special functions (Beyer 1987, pp. 426-427) include

(15) | |||

(16) | |||

(17) | |||

(18) | |||

(19) | |||

(20) | |||

(21) | |||

(22) | |||

(23) | |||

(24) |

where is the Heaviside Step Function. In general,

(25) | |||

(26) |

where the are Eulerian Numbers. Amazingly, the Z-transforms of are therefore generators for Euler's Triangle.

**References**

Beyer, W. H. (Ed.). *CRC Standard Mathematical Tables, 28th ed.* Boca Raton, FL: CRC Press, pp. 424-428, 1987.

Bracewell, R. *The Fourier Transform and Its Applications.* New York: McGraw-Hill, pp. 257-262, 1965.

© 1996-9

1999-05-26