![]() ![]() The factorials of real and imaginary numbers thus defined show uniformity in magnitude and satisfy the basic factorial equation ( c) n n ! = c( c2)( c3) … ( cn). Similarly, the author in this paper has defined the factorial function for the real negative axis. Earlier the logarithms of real negative numbers were defined on the basis of hyperbola defined for the first quadrant and extended to the negative real axis, but the author defined the logarithms for the real negative axis on the basis of hyperbola located in the third quadrant. Recently, the author (Thukral and Parkash 2014 Thukral 2014) gave a new concept on the logarithms of real negative and imaginary numbers. Later its argument was shifted down by 1, and the factorial function was extended to negative real axis and imaginary numbers. Factorial function was first defined for the positive real axis. It is seen from the historical account that the Euler’s contributions to logarithms and gamma function have revolutionized developments in science and technology (Lefort 2002 Lexa 2013). Graph for B( x ,0.5) and B(- x ,-0.5) as per the present concept. 1) have been defined for the factorials of real negative numbers and imaginary numbers.įractional factorials and multifactorials of real positive numbersįractional factorials and multifactorials of real negative numbersįractional factorials and multifactorials of imaginary positive numbersįractional factorials and multifactorials of imaginary negative numbers In the present paper, the Eularian concept of factorials has been revisited, and new functions based on Euler’s factorial function (Eqn. It is seen that till now the definition of the factorials of real negative numbers is sought from the extrapolation of gamma and other functions. There are some other factorial like products and functions, such as, primordial, double factorial, multifactorials, superfactorial, hyperfactorials etc. Ibrahim ( 2013) defined the factorial of negative integer n as the product of first n negative integers. ![]() Bhargava ( 2000) gave an expository account of the factorials, gave several new results and posed certain problems on factorials. Dutka ( 1991) gave an account of the early history of the factorial function. Mollerup, and others (Wolfram Research 2014b). The other notable contributors to the field of factorials are J. The proposed concept has also been extended to Euler’s gamma function for real negative numbers and imaginary numbers, and beta function. Fractional factorials and multifactorials have been defined in a new perspective. The moduli of the complex factorials of real negative numbers, and imaginary numbers are equal to their respective real positive number factorials. Similarly, the factorials of imaginary numbers are complex numbers. The factorials of real negative integers have their imaginary part equal to zero, thus are real numbers. As per the present concept, the factorials of real negative numbers, are complex numbers. New functions based on Euler’s factorial function have been proposed for the factorials of real negative and imaginary numbers. In the present paper, the concept of factorials has been generalised as applicable to real and imaginary numbers, and multifactorials. Presently, factorials of real negative numbers and imaginary numbers, except for zero and negative integers are interpolated using the Euler’s gamma function. ![]()
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