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All those couples of formulas are examples of the transformation $n \to n + \frac12$ removing a factor which do not depend on n. The upside-down transformation is essentially $n \to -n$ (therefore it changes $z$ to $z^{-1}$) reinterpreting $(a)_{-n}$ as $\frac{(-1)^n}{(1-a)_n}$ if $a \neq 1$, and $(1)_{-n}$ as $\frac{n(-1)^n}{(1)_n}$ which preserves formally the recurrence $\Gamma(x+1)=x \, \Gamma(x)$ (see Chapter 7 of the book A=B by Petkovsek, Wilf, Zeilberger) and another application to the WZ-method in the Section 4 of this paper). The very nice formula for $\zeta(5)$ discovered by zy_ allow us to discover a new "divergent" (convergent by analytic continuation) Ramanujan-like series for $1/\pi^4$ by using the upside-down-transformation.

The transformation $n \to n+1/2$ applied to $\lambda(n)$ essentially inverts $\lambda(n)$ giving $\frac{1}{\lambda(n)}$ but do not invert $z^n$. Hence it is not an upside-down transformation. This explains why the pattern observed in the post cannot be generalized in the way pointed out by the author.

In the Appendix of this paper there are examples of the "upside-down" technique. Another example is in the Addendum of this paper. In this unpublished file there are many examples of the transformation $n \to n + \frac12$.

All these pairs of formulas are examples of the transformation $n \to n + \frac12$ removing a factor which does not depend on n. The upside-down transformation is essentially $n \to -n$ (therefore it changes $z$ to $z^{-1}$) reinterpreting $(a)_{-n}$ as $\frac{(-1)^n}{(1-a)_n}$ if $a \neq 1$, and $(1)_{-n}$ as $\frac{n(-1)^n}{(1)_n}$ which preserves formally the recurrence $\Gamma(x+1)=x \, \Gamma(x)$ (see Chapter 7 of the book A=B by Petkovsek, Wilf, Zeilberger) and another application to the WZ-method in the Section 4 of this paper). 

The very nice formula for $\zeta(5)$ discovered by zy_ allows us to discover a new "divergent" (convergent by analytic continuation) Ramanujan-like series for $1/\pi^4$ by using the upside-down-transformation.

The transformation $n \to n+\frac12$ applied to $\lambda(n)$ essentially inverts $\lambda(n)$ giving $\frac{1}{\lambda(n)}$ but does not invert $z^n$. Hence it is not an upside-down transformation. This explains why the pattern observed in the post cannot be generalized in the way pointed out by the author.

In the Appendix of this paper there are examples of the "upside-down" technique. Another example is in the Addendum of this paper. In this unpublished file there are many examples of the transformation $n \to n + \frac12$.

All those couples of formulas are examples of the transformation $n \to n + \frac12$ removing a factor which do not depend on n. The upside-down transformation is essentially $n \to -n$ (therefore it changes $z$ to $z^{-1}$) reinterpreting $(a)_{-n}$ as $\frac{(-1)^n}{(1-a)^n}$ if $a \neq 1$, and $(1)_{-n}$ as $\frac{n(-1)^n}{(1)_n}$ which preserves formally the recurrence $\Gamma(x+1)=x \, \Gamma(x)$ (see Chapter 7 of the book A=B by Petkovsek, Wilf, Zeilberger) and another application to the WZ-method in the Section 4 of this paper). The very nice formula for $\zeta(5)$ discovered by zy_ allow us to discover a new "divergent" (convergent by analytic continuation) Ramanujan-like series for $1/\pi^4$ by using the upside-down-transformation.

The transformation $n \to n+1/2$ applied to $\lambda(n)$ essentially inverts $\lambda(n)$ giving $\frac{1}{\lambda(n)}$ but do not invert $z^n$. Hence it is not an upside-down transformation. This explains why the pattern observed in the post cannot be generalized in the way pointed out by the author.

In the Appendix of this paper there are examples of the "upside-down" technique. Another example is in the Addendum of this paper. In this unpublished file there are many examples of the transformation $n \to n + \frac12$.

All those couples of formulas are examples of the transformation $n \to n + \frac12$ removing a factor which do not depend on n. The upside-down transformation is essentially $n \to -n$ (therefore it changes $z$ to $z^{-1}$) reinterpreting $(a)_{-n}$ as $\frac{(-1)^n}{(1-a)_n}$ if $a \neq 1$, and $(1)_{-n}$ as $\frac{n(-1)^n}{(1)_n}$ which preserves formally the recurrence $\Gamma(x+1)=x \, \Gamma(x)$ (see Chapter 7 of the book A=B by Petkovsek, Wilf, Zeilberger) and another application to the WZ-method in the Section 4 of this paper). The very nice formula for $\zeta(5)$ discovered by zy_ allow us to discover a new "divergent" (convergent by analytic continuation) Ramanujan-like series for $1/\pi^4$ by using the upside-down-transformation.

The transformation $n \to n+1/2$ applied to $\lambda(n)$ essentially inverts $\lambda(n)$ giving $\frac{1}{\lambda(n)}$ but do not invert $z^n$. Hence it is not an upside-down transformation. This explains why the pattern observed in the post cannot be generalized in the way pointed out by the author.

In the Appendix of this paper there are examples of the "upside-down" technique. Another example is in the Addendum of this paper. In this unpublished file there are many examples of the transformation $n \to n + \frac12$.

All those couples of formulas are examples of the transformation $n \to n + \frac12$ removing a factor which do not depends on n. The upside-down transformation is essentially $n \to -n$ (therefore it changes $z$ to $z^{-1}$) reinterpreting $(a)_{-n}$ as $\frac{(-1)^n}{(1-a)^n}$ if $a \neq 1$, and $(1)_{-n}$ as $\frac{n(-1)^n}{(1)_n}$ which preserves formally the recurrence $\Gamma(x+1)=x \, \Gamma(x)$ (see Chapter 7 of the book A=B by Petkovsek, Wilf, Zeilberger) and another application to the WZ-method in the Section 4 of this paper). The very nice formula for $\zeta(5)$ discovered by zy_ allow us to discover a new "divergent" (convergent by analytic continuation) Ramanujan-like series for $1/\pi^4$ by using the upside-down-transformation.

The transformation $n \to n+1/2$ applied to $\lambda(n)$ essentially inverts $\lambda(n)$ giving $\frac{1}{\lambda(n)}$ but do not invert $z^n$. Hence it is not an upside-down transformation. This explains why the pattern observed in the post cannot be generalized in the way pointed out by the author.

In the Appendix of this paper there are examples of the "upside-down" technique and in this unpublished file there are many examples of the transformation $n \to n + \frac12$.

All those couples of formulas are examples of the transformation $n \to n + \frac12$ removing a factor which do not depend on n. The upside-down transformation is essentially $n \to -n$ (therefore it changes $z$ to $z^{-1}$) reinterpreting $(a)_{-n}$ as $\frac{(-1)^n}{(1-a)^n}$ if $a \neq 1$, and $(1)_{-n}$ as $\frac{n(-1)^n}{(1)_n}$ which preserves formally the recurrence $\Gamma(x+1)=x \, \Gamma(x)$ (see Chapter 7 of the book A=B by Petkovsek, Wilf, Zeilberger) and another application to the WZ-method in the Section 4 of this paper). The very nice formula for $\zeta(5)$ discovered by zy_ allow us to discover a new "divergent" (convergent by analytic continuation) Ramanujan-like series for $1/\pi^4$ by using the upside-down-transformation.

The transformation $n \to n+1/2$ applied to $\lambda(n)$ essentially inverts $\lambda(n)$ giving $\frac{1}{\lambda(n)}$ but do not invert $z^n$. Hence it is not an upside-down transformation. This explains why the pattern observed in the post cannot be generalized in the way pointed out by the author.

In the Appendix of this paper there are examples of the "upside-down" technique. Another example is in the Addendum of this paper. In this unpublished file there are many examples of the transformation $n \to n + \frac12$.