The numbers , where is a Hermite polynomial, may be called Hermite numbers. For , 1, ..., the first few are 1, 0, , 0, 12, 0, , 0, 1680, 0, ... (OEIS A067994). They are given explicitly by(1)(2)As a result of the ratio always being divisible by for , the only prime Hermite number is .The Hermite numbers are related to the Hermite polynomials by(3)where , and(4)where .
The tribonacci numbers are a generalization of the Fibonacci numbers defined by , , , and the recurrence equation(1)for (e.g., Develin 2000). They represent the case of the Fibonacci n-step numbers.The first few terms using the above indexing convention for , 1, 2, ... are 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, ... (OEIS A000073; which however adopts the alternate indexing convention and ).The first few prime tribonacci numbers are 2, 7, 13, 149, 19341322569415713958901, ... (OEIS A092836), which have indices 3, 5, 6, 10, 86, 97, 214, 801, 4201, 18698, 96878, ... (OEIS A092835), and no others with (E. W. Weisstein, Mar. 21, 2009).Using Brown's criterion, it can be shown that the tribonacci numbers are complete; that is, every positive number can be written as the sum of distinct tribonacci numbers. Moreover, every positive number has a unique Zeckendorf-like expansion as the sum of distinct tribonacci numbers and that sum does..
The cosine function is one of the basic functions encountered in trigonometry (the others being the cosecant, cotangent, secant, sine, and tangent). Let be an angle measured counterclockwise from the x-axis along the arc of the unit circle. Then is the horizontal coordinate of the arc endpoint.The common schoolbook definition of the cosine of an angle in a right triangle (which is equivalent to the definition just given) is as the ratio of the lengths of the side of the triangle adjacent to the angle and the hypotenuse, i.e.,(1)A convenient mnemonic for remembering the definition of the sine, cosine, and tangent is SOHCAHTOA (sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, tangent equals opposite over adjacent).As a result of its definition, the cosine function is periodic with period . By the Pythagorean theorem, also obeys the identity(2) Min Max Re Im The definition of the cosine function can be extended to..
The sine function is one of the basic functions encountered in trigonometry (the others being the cosecant, cosine, cotangent, secant, and tangent). Let be an angle measured counterclockwise from the x-axis along an arc of the unit circle. Then is the vertical coordinate of the arc endpoint, as illustrated in the left figure above.The common schoolbook definition of the sine of an angle in a right triangle (which is equivalent to the definition just given) is as the ratio of the lengths of the side of the triangle opposite the angle and the hypotenuse, i.e.,(1)A convenient mnemonic for remembering the definition of the sine, as well as the cosine and tangent, is SOHCAHTOA (sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, tangent equals opposite over adjacent).As a result of its definition, the sine function is periodic with period . By the Pythagorean theorem, also obeys the identity(2) Min Max Re Im The definition..
The Hermite polynomials are set of orthogonal polynomials over the domain with weighting function , illustrated above for , 2, 3, and 4. Hermite polynomials are implemented in the Wolfram Language as HermiteH[n, x].The Hermite polynomial can be defined by the contour integral(1)where the contour encloses the origin and is traversed in a counterclockwise direction (Arfken 1985, p. 416).The first few Hermite polynomials are(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)When ordered from smallest to largest powers, the triangle of nonzero coefficientsis 1; 2; -2, 4; -12, 8; 12, -48, 16; 120, -160, 32; ... (OEIS A059343).The values may be called Hermite numbers.The Hermite polynomials are a Sheffer sequencewith(13)(14)(Roman 1984, p. 30), giving the exponentialgenerating function(15)Using a Taylor series shows that(16)(17)Since ,(18)(19)Now define operators(20)(21)It follows that(22)(23)(24)(25)(26)so(27)and(28)(Arfken..
The Riemann zeta function is an extremely important special function of mathematics and physics that arises in definite integration and is intimately related with very deep results surrounding the prime number theorem. While many of the properties of this function have been investigated, there remain important fundamental conjectures (most notably the Riemann hypothesis) that remain unproved to this day. The Riemann zeta function is denoted and is plotted above (using two different scales) along the real axis. Min Max Re Im In general, is defined over the complex plane for one complex variable, which is conventionally denoted (instead of the usual ) in deference to the notation used by Riemann in his 1859 paper that founded the study of this function (Riemann 1859). is implemented in the Wolfram Language as Zeta[s].The plot above shows the "ridges" of for and . The fact that the ridges appear to decrease monotonically for is not..